Radial Delta Function, and the 3-D Fourier-Bessel - Gauge-institute ...
Radial Delta Function, and the 3-D Fourier-Bessel - Gauge-institute ...
Radial Delta Function, and the 3-D Fourier-Bessel - Gauge-institute ...
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
<strong>Radial</strong> <strong>Delta</strong> <strong>Function</strong> <strong>and</strong> <strong>the</strong><br />
3-D <strong>Fourier</strong>-<strong>Bessel</strong> Transform<br />
H. Vic Dannon<br />
vic0@comcast.net<br />
October, 2010<br />
Abstract In [Dan6], we have shown that<br />
δ(,, xyz) = δ()()()<br />
xδ<br />
yδ<br />
z<br />
is <strong>the</strong> 3-dimensional <strong>Fourier</strong> Transform of <strong>the</strong> function 1. Here,<br />
we show that its <strong>Radial</strong> form is <strong>the</strong> 3-dimensional <strong>Fourier</strong>-<strong>Bessel</strong><br />
Transform of <strong>the</strong> function 1,<br />
ν=∞<br />
ν=<br />
0<br />
( πνr<br />
)<br />
sin 2<br />
δ( x) δ( y) δ( z) = 4∫ νdν,<br />
r<br />
2 2<br />
r = x + y + z<br />
2 .<br />
Thus,<br />
δ( x) δ( y) δ( z)<br />
is a <strong>Radial</strong>ly Symmetric <strong>Delta</strong> <strong>Function</strong> that we<br />
will denote δ r .<br />
Similarly,<br />
<strong>Radial</strong> ()<br />
ν=∞<br />
2 1<br />
δ( x −ξ) δ( y −η) δ( z − ζ) = 4 sin( 2πνr) sin( 2πνσ)<br />
rσ<br />
∫ dν<br />
ν=<br />
0<br />
is <strong>Radial</strong>ly symmetric, <strong>and</strong> defines <strong>the</strong> <strong>Radial</strong> <strong>Delta</strong> δ <strong>Radial</strong> ( r − σ)<br />
.<br />
The formulas above are exclusively Hyper-real. They cannot be<br />
derived by means of <strong>the</strong> Calculus of Limits, <strong>and</strong> are unknown in<br />
<strong>the</strong> Calculus of Limits.<br />
1
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
The <strong>Radial</strong> <strong>Delta</strong> <strong>Function</strong><br />
δ <strong>Radial</strong> (r − σ)<br />
is a Discontinuous,<br />
Hyper-real function, that spikes at r = σ , <strong>and</strong> vanishes for σ ≠ r .<br />
The <strong>Fourier</strong>-<strong>Bessel</strong> Integral Theorem for <strong>Radial</strong>ly symmetric<br />
function f () r guarantees that <strong>the</strong> 3-dimesional <strong>Fourier</strong>-<strong>Bessel</strong><br />
Transform <strong>and</strong> its Inverse are well defined operations, so that<br />
inversion yields <strong>the</strong> original function that generated <strong>the</strong><br />
Transform.<br />
It is believed to hold in <strong>the</strong> Calculus of Limits under given<br />
conditions. In fact, it does not hold in <strong>the</strong> Calculus of Limits<br />
because <strong>the</strong> integration of <strong>the</strong> <strong>Fourier</strong> Integral requires <strong>the</strong><br />
ν=∞<br />
∫<br />
( ) ( )<br />
integration of sin 2πνσ sin 2πνr dν<br />
that diverges at σ = r .<br />
ν=<br />
0<br />
Only in Infinitesimal Calculus, we can integrate over<br />
singularities, <strong>and</strong> <strong>the</strong> <strong>Fourier</strong> Integral Theorem holds.<br />
⎛<br />
⎞ ⎟<br />
f () r = 4 ⎜4 f()<br />
d ⎟<br />
⎝<br />
⎠<br />
( πνσ ) ( πνr<br />
)<br />
ν=∞ σ=∞<br />
sin 2 sin 2<br />
⎜<br />
2 2<br />
σ σ σ<br />
∫ ∫ ν d<br />
νσ<br />
νr<br />
ν= 0⎜<br />
σ=<br />
0<br />
⎟<br />
σ=∞ ⎛ ν=∞<br />
⎞ 2 1<br />
2<br />
= f ( σ) 4 sin( 2πνσ) sin( 2πνr)<br />
d d<br />
rσ<br />
σ= 0<br />
⎜<br />
⎝ ν=<br />
0<br />
⎠⎟<br />
∫ ∫ ν σ σ<br />
σ=∞ ⎛ ν=∞<br />
⎞ 2<br />
= f ( σ) 4 sin( 2πνσ) sin( 2πνr)<br />
d d<br />
σ= 0<br />
⎜<br />
⎝ ν=<br />
0<br />
⎠⎟<br />
∫ ∫ ν σ.<br />
Keywords: Infinitesimal, Infinite-Hyper-real, Hyper-real,<br />
ν<br />
2
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
Cardinal, Infinity. Non-Archimedean, Non-St<strong>and</strong>ard Analysis,<br />
Calculus, Limit, Continuity, Derivative, Integral,<br />
2000 Ma<strong>the</strong>matics Subject Classification 26E35; 26E30;<br />
26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15;<br />
46S20; 97I40; 97I30.<br />
Contents<br />
Introduction<br />
1. Hyper-real Line<br />
2. Hyper-real Integral<br />
3. <strong>Delta</strong> <strong>Function</strong><br />
4. The <strong>Fourier</strong> Transform<br />
δ( r −r<br />
) δ( θ −θ ) δ(<br />
φ−φ<br />
0<br />
)<br />
5.<br />
0 0<br />
6. <strong>Radial</strong> <strong>Delta</strong> δ <strong>Radial</strong><br />
() r<br />
7. <strong>Radial</strong> <strong>Delta</strong> δ ( − σ)<br />
<strong>Radial</strong><br />
r<br />
8. δ<strong>Radial</strong>(<br />
r − r0<br />
) <strong>and</strong> δ( r − r0<br />
)<br />
9. 3-D <strong>Fourier</strong>-<strong>Bessel</strong> Transform<br />
10. <strong>Fourier</strong>-<strong>Bessel</strong> Integral Theorem holds only in Infinitesimal<br />
Calculus<br />
References<br />
3
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
Introduction<br />
It is believed that a radially symmetric f () r , satisfies <strong>the</strong> 3-<br />
dimensional <strong>Fourier</strong>-<strong>Bessel</strong> Integral Theorem<br />
⎛<br />
⎞ ⎟<br />
f () r = 4 ⎜4 f()<br />
d ⎟<br />
⎝<br />
⎠<br />
( πνσ ) ( πνr<br />
)<br />
ν=∞ σ=∞<br />
sin 2 sin 2<br />
⎜<br />
2 2<br />
σ σ σ<br />
∫ ∫ ν d<br />
νσ<br />
νr<br />
ν= 0⎜<br />
σ=<br />
0<br />
⎟<br />
σ=∞ ⎛ ν=∞<br />
⎞ 2 1<br />
2<br />
= f ( σ) 4 sin( 2πνσ) sin( 2πνr)<br />
d d<br />
rσ<br />
σ= 0<br />
⎜<br />
⎝ ν=<br />
0<br />
⎠⎟<br />
∫ ∫ ν σ σ<br />
However, in <strong>the</strong> Calculus of Limits, at σ = r ,<br />
ν=∞<br />
ν=<br />
0<br />
( ) ( )<br />
∫ sin 2πνσ sin 2πνr dν<br />
=∞.<br />
ν<br />
Avoiding <strong>the</strong> singularity at<br />
σ = r<br />
does not recover <strong>the</strong> <strong>Fourier</strong>-<br />
<strong>Bessel</strong> Integral Theorem, because without <strong>the</strong> singularity <strong>the</strong><br />
integral equals zero.<br />
Thus, <strong>the</strong> <strong>Fourier</strong> Integral Theorem cannot be written in <strong>the</strong><br />
Calculus of Limits.<br />
In Infinitesimal Calculus [Dan4], <strong>the</strong> singularity can be integrated<br />
over, <strong>and</strong> defines <strong>the</strong> Spherical <strong>Delta</strong> <strong>Function</strong> δ <strong>Radial</strong> ( r − σ)<br />
.<br />
Then, for any Hyper-real function f () r , <strong>the</strong> <strong>Fourier</strong>-<strong>Bessel</strong><br />
Integral Theorem is <strong>the</strong> sifting property of δ <strong>Radial</strong> ( r − σ)<br />
.<br />
In <strong>the</strong> calculus of limits,<br />
δ <strong>Radial</strong> (r − σ)<br />
cannot be defined.<br />
4
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
1.<br />
Hyper-real Line<br />
Each real number α can be represented by a Cauchy sequence of<br />
rational numbers, ( r , r , r ,...) so that r → α .<br />
1 2 3<br />
The constant sequence ( ααα , , ,...) is a constant Hyper-real.<br />
In [Dan2] we established that,<br />
1. Any totally ordered set of positive, monotonically decreasing<br />
n<br />
to zero sequences<br />
infinitesimal Hyper-reals.<br />
( ι1, ι2, ι3,...)<br />
constitutes a family of<br />
2. The infinitesimals are smaller than any real number, yet<br />
strictly greater than zero.<br />
1 1 1<br />
3. Their reciprocals ( , , ,...<br />
ι 1<br />
ι 2<br />
ι 3<br />
) are <strong>the</strong> infinite Hyper-reals.<br />
4. The infinite Hyper-reals are greater than any real number,<br />
yet strictly smaller than infinity.<br />
5. The infinite Hyper-reals with negative signs are smaller<br />
than any real number, yet strictly greater than −∞.<br />
6. The sum of a real number with an infinitesimal is a<br />
non-constant Hyper-real.<br />
7. The Hyper-reals are <strong>the</strong> totality of constant Hyper-reals, a<br />
family of infinitesimals, a family of infinitesimals with<br />
5
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
negative sign, a family of infinite Hyper-reals, a family of<br />
infinite Hyper-reals with negative sign, <strong>and</strong> non-constant<br />
Hyper-reals.<br />
8. The Hyper-reals are totally ordered, <strong>and</strong> aligned along a<br />
line: <strong>the</strong> Hyper-real Line.<br />
9. That line includes <strong>the</strong> real numbers separated by <strong>the</strong> nonconstant<br />
Hyper-reals. Each real number is <strong>the</strong> center of an<br />
interval of Hyper-reals, that includes no o<strong>the</strong>r real number.<br />
10. In particular, zero is separated from any positive real<br />
by <strong>the</strong> infinitesimals, <strong>and</strong> from any negative real by <strong>the</strong><br />
infinitesimals with negative signs, −dx .<br />
11. Zero is not an infinitesimal, because zero is not strictly<br />
greater than zero.<br />
12. We do not add infinity to <strong>the</strong> Hyper-real line.<br />
13. The infinitesimals, <strong>the</strong> infinitesimals with negative<br />
signs, <strong>the</strong> infinite Hyper-reals, <strong>and</strong> <strong>the</strong> infinite Hyper-reals<br />
with negative signs are semi-groups with<br />
respect to addition. Nei<strong>the</strong>r set includes zero.<br />
14. The Hyper-real line is embedded in , <strong>and</strong> is not<br />
∞<br />
homeomorphic to <strong>the</strong> real line. There is no bi-continuous<br />
one-one mapping from <strong>the</strong> Hyper-real onto <strong>the</strong> real line.<br />
6
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
15. In particular, <strong>the</strong>re are no points on <strong>the</strong> real line that<br />
can be assigned uniquely to <strong>the</strong> infinitesimal Hyper-reals, or<br />
to <strong>the</strong> infinite Hyper-reals, or to <strong>the</strong> non-constant Hyperreals.<br />
16. No neighbourhood of a Hyper-real is homeomorphic to<br />
an<br />
n<br />
ball. Therefore, <strong>the</strong> Hyper-real line is not a manifold.<br />
17. The Hyper-real line is totally ordered like a line, but it<br />
is not spanned by one element, <strong>and</strong> it is not one-dimensional.<br />
7
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
2.<br />
Hyper-real Integral<br />
In [Dan3], we defined <strong>the</strong> Hyper-real integral of a Hyper-real<br />
<strong>Function</strong>.<br />
Let f () x be a Hyper-real function on <strong>the</strong> interval [ ab] , .<br />
The interval may not be bounded.<br />
f () x may take infinite Hyper-real values, <strong>and</strong> need not be<br />
bounded.<br />
At each<br />
a<br />
≤<br />
x<br />
≤b,<br />
<strong>the</strong>re is a rectangle with base<br />
dx dx<br />
[ x − , x + 2<br />
], height f () x , <strong>and</strong> area<br />
2<br />
f ( xdx. )<br />
We form <strong>the</strong> Integration Sum of all <strong>the</strong> areas for <strong>the</strong> x ’s that<br />
start at x = a, <strong>and</strong> end at x = b,<br />
∑ f ( xdx ) .<br />
x∈[ a, b]<br />
If for any infinitesimal dx , <strong>the</strong> Integration Sum has <strong>the</strong> same<br />
Hyper-real value, <strong>the</strong>n f () x is integrable over <strong>the</strong> interval [ ab] , .<br />
Then, we call <strong>the</strong> Integration Sum <strong>the</strong> integral of f () x from x = a,<br />
to x<br />
= b, <strong>and</strong> denote it by<br />
8
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
x=<br />
b<br />
∫ f ( xdx ) .<br />
x=<br />
a<br />
If <strong>the</strong> Hyper-real is infinite, <strong>the</strong>n it is <strong>the</strong> integral over [, ab] ,<br />
If <strong>the</strong> Hyper-real is finite,<br />
x=<br />
b<br />
∫ fxdx ( ) = real part of <strong>the</strong> hyper-real . <br />
x=<br />
a<br />
2.1 The countability of <strong>the</strong> Integration Sum<br />
In [Dan1], we established <strong>the</strong> equality of all positive infinities:<br />
We proved that <strong>the</strong> number of <strong>the</strong> Natural Numbers,<br />
Card , equals <strong>the</strong> number of Real Numbers,<br />
2 Card <br />
Card = , <strong>and</strong><br />
we have<br />
2 Card<br />
2<br />
Card <br />
Card = ( Card) = .... = 2 = 2 = ... ≡ ∞.<br />
In particular, we demonstrated that <strong>the</strong> real numbers may be<br />
well-ordered.<br />
Consequently, <strong>the</strong>re are countably many real numbers in <strong>the</strong><br />
interval [ ab] , , <strong>and</strong> <strong>the</strong> Integration Sum has countably many terms.<br />
While we do not sequence <strong>the</strong> real numbers in <strong>the</strong> interval, <strong>the</strong><br />
summation takes place over countably many f ( xdx. )<br />
9
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
The Lower Integral is <strong>the</strong> Integration Sum where f ( x ) is replaced<br />
by its lowest value on each interval<br />
2.2<br />
∑<br />
x∈[ a, b]<br />
⎛<br />
⎜⎝<br />
dx dx<br />
2 2<br />
[ x − , x + ]<br />
⎞<br />
inf f ( t)<br />
dx<br />
⎠⎟<br />
x− ≤t≤ x+<br />
dx dx<br />
2 2<br />
The Upper Integral is <strong>the</strong> Integration Sum where f ( x ) is replaced<br />
by its largest value on each interval<br />
2.3<br />
∑<br />
x∈[ a, b]<br />
⎛<br />
⎜⎝<br />
dx dx<br />
2 2<br />
[ x − , x + ]<br />
⎞ sup f ( t)<br />
dx<br />
⎠⎟<br />
x− ≤t≤ x+<br />
dx dx<br />
2 2<br />
If <strong>the</strong> integral is a finite Hyper-real, we have<br />
2.4 A Hyper-real function has a finite integral if <strong>and</strong> only if its<br />
upper integral <strong>and</strong> its lower integral are finite, <strong>and</strong> differ by an<br />
infinitesimal.<br />
10
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
3.<br />
<strong>Delta</strong> <strong>Function</strong><br />
In [Dan5], we defined <strong>the</strong> <strong>Delta</strong> <strong>Function</strong>, <strong>and</strong> established its<br />
properties<br />
1. The <strong>Delta</strong> <strong>Function</strong> is a Hyper-real function defined from <strong>the</strong><br />
Hyper-real line into <strong>the</strong> set of two Hyper-reals<br />
⎧<br />
⎪ 1 ⎫<br />
⎨0, ⎪<br />
⎬<br />
⎪⎩<br />
dx<br />
⎭⎪ . The<br />
Hyper-real 0 is <strong>the</strong> sequence<br />
0, 0, 0,... . The infinite Hyperreal<br />
1<br />
dx<br />
depends on our choice of dx .<br />
2. We will usually choose <strong>the</strong> family of infinitesimals that is<br />
spanned by <strong>the</strong> sequences<br />
1<br />
n , 1<br />
2<br />
n<br />
,<br />
1<br />
n<br />
3<br />
,… It is a<br />
semigroup with respect to vector addition, <strong>and</strong> includes all<br />
<strong>the</strong> scalar multiples of <strong>the</strong> generating sequences that are<br />
non-zero. That is, <strong>the</strong> family includes infinitesimals with<br />
negative sign. Therefore,<br />
1<br />
dx<br />
will mean <strong>the</strong> sequence n .<br />
Alternatively, we may choose <strong>the</strong> family spanned by <strong>the</strong><br />
sequences<br />
1<br />
2 n ,<br />
1<br />
3 n ,<br />
1<br />
4 n ,… Then, 1<br />
dx<br />
will mean <strong>the</strong><br />
11
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
sequence 2 n<br />
. Once we determined <strong>the</strong> basic infinitesimal<br />
dx , we will use it in <strong>the</strong> Infinite Riemann Sum that defines<br />
an Integral in Infinitesimal Calculus.<br />
3. The <strong>Delta</strong> <strong>Function</strong> is strictly smaller than ∞<br />
4. We define,<br />
1<br />
χ δ ( x) ≡ dx ( )<br />
,<br />
dx x<br />
dx<br />
⎡ ⎤ ,<br />
⎢−<br />
⎣ 2 2 ⎥⎦<br />
where<br />
χ ⎡<br />
⎢−<br />
⎣<br />
dx,<br />
dx<br />
2 2<br />
⎧ dx dx<br />
1, x ∈ ⎡−<br />
, ⎤<br />
( x)<br />
= ⎪ ⎢ 2 2 ⎥<br />
⎨ ⎣ ⎦ .<br />
⎪⎪ 0, o<strong>the</strong>rwise<br />
⎩<br />
⎤<br />
⎥⎦<br />
5. Hence,<br />
for x < 0 , δ ( x) = 0<br />
at<br />
for<br />
dx<br />
x =− , δ( x)<br />
jumps from 0 to<br />
2<br />
dx dx<br />
⎢ ⎣<br />
,<br />
2 2 ⎥ ⎦ , 1<br />
( x)<br />
x ∈ ⎡−<br />
⎤<br />
δ = .<br />
dx<br />
1<br />
dx ,<br />
at x = 0 ,<br />
δ (0) =<br />
1<br />
dx<br />
at<br />
dx<br />
x = , δ( x)<br />
drops from<br />
2<br />
for x > 0 , δ ( x) = 0.<br />
xδ ( x) = 0<br />
1<br />
dx to 0.<br />
6. If dx =<br />
1<br />
, ( x) = 1 1( x),2 1 1( x),3 1 1( x )...<br />
n<br />
[ − , ] [ − , ] [ − , ]<br />
δ χ χ χ<br />
2 2 4 4 6 6<br />
7. If dx =<br />
2<br />
,<br />
n<br />
1 2 3<br />
δ ( x) = , , ,...<br />
2 2 2<br />
2 cosh x 2 cosh 2x 2 cosh 3x<br />
12
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
8. If dx =<br />
1<br />
,<br />
n<br />
− x − 2x − 3x<br />
[0, ∞) [0, ∞) [0, ∞)<br />
δ( x) = e χ ,2 e χ , 3 e χ ,...<br />
x =∞<br />
∫<br />
9. δ( xdx ) = 1.<br />
x =−∞<br />
13
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
4.<br />
The <strong>Fourier</strong> Transform<br />
In [Dan6], we defined <strong>the</strong> <strong>Fourier</strong> Transform <strong>and</strong> established its<br />
properties<br />
1. F { x }<br />
δ ( ) = 1<br />
2. δ( x)<br />
= <strong>the</strong> inverse <strong>Fourier</strong> Transform of <strong>the</strong> unit function 1<br />
ω=∞<br />
1<br />
=<br />
2π<br />
∫<br />
ν=∞<br />
ω=−∞<br />
e<br />
2πix<br />
iωx<br />
dω<br />
= ∫ e dν<br />
, ω = 2πν<br />
ν=−∞<br />
3.<br />
1<br />
2π<br />
ω=∞<br />
∫<br />
ω=−∞<br />
e<br />
iωx<br />
dω<br />
x = 0<br />
=<br />
1 = an infinite Hyper-real<br />
dx<br />
ω=∞<br />
∫<br />
ω=−∞<br />
e<br />
iωx<br />
dω<br />
x ≠0<br />
=<br />
0<br />
4. <strong>Fourier</strong> Integral Theorem<br />
k =∞ ⎛ ξ=∞<br />
⎞<br />
1<br />
−ikξ<br />
f ( x) = f( ξ)<br />
e ξ<br />
2π<br />
k =−∞ ⎜<br />
⎝ξ=−∞<br />
⎠⎟<br />
ikx<br />
∫ ∫ d e dk<br />
does not hold in <strong>the</strong> Calculus of Limits, under any<br />
conditions.<br />
14
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
5. <strong>Fourier</strong> Integral Theorem in Infinitesimal Calculus<br />
If f ( x ) is Hyper-real function,<br />
Then,<br />
<strong>the</strong> Hyper-real <strong>Fourier</strong> Integral Theorem holds.<br />
x =∞<br />
−iαx<br />
∫ f ( xe ) dx converges to F( α)<br />
x =−∞<br />
<br />
1<br />
2π<br />
α=∞<br />
−iαx<br />
∫ F( α)<br />
e dα converges to f ( x )<br />
α=−∞<br />
6. 3-Dimesional <strong>Fourier</strong> Transform<br />
F<br />
{ }<br />
z=∞ y=∞ x=∞<br />
−iω x−iω y−iω<br />
z<br />
x y z<br />
f (,, xyz) fxyze (,, ) dxdydz<br />
= ∫ ∫ ∫<br />
z=−∞ y=−∞ x=−∞<br />
z=∞ y=∞ x=∞<br />
− 2 πi( ν x+ ν y+<br />
ν z)<br />
x y z<br />
= ∫ ∫ ∫ f (,, x y z) e<br />
dxdydz ,<br />
z=−∞ y=−∞ x=−∞<br />
ω<br />
ω<br />
ω<br />
x<br />
y<br />
z<br />
= 2πν<br />
= 2πν<br />
= 2πν<br />
x<br />
y<br />
z<br />
8. 3-Dimesional Inverse <strong>Fourier</strong> Transform<br />
−1<br />
ωz=∞ ωy<br />
=∞ ωx=∞<br />
1<br />
i( ωxx+ ωyy+<br />
ωzz)<br />
x y z 3<br />
x y z x y z<br />
(2 π)<br />
ω =−∞ ω =−∞ ω =−∞<br />
{ } = ∫ ∫ ∫<br />
F F( ω , ω , ω ) F( ω , ω , ω ) e<br />
dω dω dω<br />
z y x<br />
ν =∞ ν =∞ ν =∞<br />
z<br />
∫ ∫ ∫<br />
2 πi( νxx+ νyy+<br />
νzz)<br />
= F(2 πν ,2 πν ,2 πν ) e dν dν dν<br />
,<br />
ν =−∞ ν =−∞ ν =−∞<br />
z y x<br />
y<br />
x<br />
x y z x y<br />
z<br />
ω<br />
ω<br />
ω<br />
x<br />
y<br />
z<br />
= 2πν<br />
= 2πν<br />
= 2πν<br />
x<br />
y<br />
z<br />
15
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
9. 3-Dimesional <strong>Fourier</strong> Integral Theorem<br />
ωz=∞ ωy<br />
=∞ ωx=∞ ⎛ ζ=∞ η=∞ ξ=∞<br />
⎞<br />
1<br />
−iωξ x<br />
−iωη y<br />
−iωζ<br />
z<br />
fxyz (,, ) = f(, ξηζ , ) e ddd ξ ηζ<br />
3 ∫ ∫ ∫ ×<br />
(2 π)<br />
∫ ∫ ∫<br />
ω =−∞ ω =−∞ ω =−∞<br />
⎜⎝ζ=−∞ η=−∞ ξ=−∞<br />
⎠⎟<br />
z y x<br />
x<br />
i( ω x+ iω y+<br />
iω<br />
z)<br />
x y z<br />
× e d d d<br />
y<br />
ω ω ω<br />
x y z<br />
ζ=∞ η=∞ ξ=∞ ⎛ ω<br />
1 x =∞<br />
⎞ ⎛ ωy<br />
=∞<br />
⎞<br />
iω<br />
( ) 1<br />
i<br />
y( y )<br />
( , , )<br />
x<br />
x−ξ<br />
ω −η<br />
= ∫ ∫ ∫ f ξηζ e dω 2<br />
x<br />
dξ e dω 2<br />
y<br />
η×<br />
π<br />
∫ π<br />
∫ ζ=−∞ η=−∞ ξ=−∞ ⎜ d<br />
⎝ ω =−∞ ⎠⎟ ⎜⎝ ω =−∞<br />
⎠⎟<br />
⎛ ωz<br />
=∞<br />
⎞ 1<br />
iωz<br />
( z−ζ)<br />
e dω<br />
×⎜<br />
⎜<br />
z<br />
ζ<br />
2π<br />
∫<br />
d<br />
⎜<br />
⎝ ω =−∞<br />
⎠⎟<br />
ζ=∞ η=∞ ξ=∞ ⎛ νx<br />
=∞<br />
⎞ ⎛ νy<br />
=∞<br />
⎞<br />
2 πν i ( )<br />
2 i<br />
y( y )<br />
( , , )<br />
x<br />
x−ξ<br />
πν −η<br />
= ∫ ∫ ∫ f ξηζ e dν x<br />
dξ e dν y<br />
dη×<br />
∫ ∫<br />
ζ=−∞ η=−∞ ξ=−∞ ⎝⎜ ν =−∞ ⎠⎟ ⎜⎝ν<br />
=−∞<br />
⎠⎟<br />
x<br />
y<br />
z<br />
10. 3-Dimesional <strong>Delta</strong> <strong>Function</strong><br />
⎛ νz<br />
=∞<br />
⎞ 2 πν i<br />
z<br />
( z−ζ)<br />
e dν<br />
×⎜<br />
⎜<br />
z<br />
dζ<br />
∫ ,<br />
⎜⎝ν<br />
=−∞<br />
⎠⎟<br />
z<br />
ω<br />
ω<br />
ω<br />
x<br />
y<br />
z<br />
= 2πν<br />
= 2πν<br />
= 2πν<br />
x<br />
y<br />
z<br />
⎛ ωx<br />
=∞ ⎞⎛<br />
ωy<br />
=∞ ⎞⎛<br />
ωz<br />
=∞<br />
1 i 1 i 1<br />
yy<br />
( xyz , ,<br />
xx i<br />
zz<br />
)<br />
x<br />
y<br />
2 e ω<br />
d 2 e ω<br />
d 2<br />
e ω<br />
δ = ω ω<br />
π π π<br />
d ω<br />
⎝⎜<br />
ω =−∞ ⎠⎟<br />
⎜⎝<br />
ω =−∞ ⎠⎟<br />
⎝⎜<br />
ω =−∞<br />
∫ ∫ ∫ z<br />
x y z<br />
⎞<br />
⎠⎟<br />
⎛ ν<br />
ν =∞<br />
x=∞ ⎞⎛<br />
y<br />
⎞⎛<br />
νz=∞<br />
πν<br />
πν<br />
2 i πν<br />
e dν<br />
2<br />
=<br />
xx i<br />
yy<br />
e dν<br />
2 i e<br />
zz<br />
d<br />
∫<br />
⎜<br />
∫ ∫ ν<br />
⎝ν =−∞ ⎠⎟⎜<br />
⎝ν =−∞ ⎠⎟<br />
⎝⎜ν<br />
=−∞<br />
x y z<br />
x y z<br />
⎞<br />
,<br />
⎠⎟<br />
ω<br />
ω<br />
ω<br />
x<br />
y<br />
z<br />
= 2πν<br />
= 2πν<br />
= 2πν<br />
x<br />
y<br />
z<br />
16
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
5.<br />
δ( r −r<br />
) δ( θ −θ ) δ(<br />
φ −φ<br />
0<br />
)<br />
0 0<br />
5.1<br />
1<br />
δ ( r − r0 ) = χ[ dr<br />
0 , dr<br />
2 0<br />
] ( )<br />
r − r +<br />
dr r , r ≥ 0<br />
2<br />
5.2<br />
1<br />
δθ ( − θ<br />
0 ) = χ[ dθ<br />
0 , dθ<br />
2 0<br />
] ( θ)<br />
, 0 ≤ θ ≤ π<br />
θ − θ +<br />
dθ 2<br />
5.3<br />
1<br />
δφ ( − φ<br />
0 ) = χ[ dφ<br />
( )<br />
0 , dφ<br />
φ,<br />
0 ≤ φ ≤ 2π<br />
dφ φ − φ<br />
2 0+<br />
] 2<br />
The product<br />
δ( r −r<br />
) δ( θ −θ ) δ(<br />
φ−<br />
0 0<br />
0 )<br />
φ 0 )<br />
defines a <strong>Delta</strong> <strong>Function</strong><br />
that sifts along a line through its singularity at ( r , θ , φ ).<br />
0 0 0<br />
5.4<br />
0 0 0 0 0<br />
δ( r −r , θ −θ , φ−φ ) ≡ δ( r −r<br />
) δ( θ −θ ) δ(<br />
φ−<br />
φ<br />
=<br />
1 χ<br />
1 1<br />
[ dr<br />
0 , dr<br />
2 0 ] () r χ[<br />
2 0 , 2 0 ] () ()<br />
r r<br />
dθ dθ θ χ<br />
2 [ dφ 0 , dφ<br />
φ<br />
− + θ − θ +<br />
dr dθ<br />
dφ φ − φ<br />
2 0+<br />
] 2<br />
Transforming between Spherical <strong>and</strong> Cartesian Coordinates<br />
17
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
x<br />
y<br />
z<br />
= r sin θcosφ<br />
= r sin θsin<br />
φ , y = r sin θ sin φ ,<br />
= r cos θ<br />
x<br />
z<br />
= r<br />
sin θ<br />
0 0 0<br />
0 0 0<br />
= r<br />
cosθ<br />
0 0 0<br />
cosφ<br />
0<br />
0<br />
r<br />
0<br />
> 0<br />
1<br />
δ( − ) δ( − ) δ( − ) = δ( − ) δ( θ −θ ) δ(<br />
φ−φ<br />
0<br />
)<br />
5.5 x x0 y y0 z z0 r r<br />
2<br />
0 0<br />
r0 sin θ0<br />
Proof:<br />
δ( r −r ) δ( θ −θ ) δ( φ− φ ) drdθdφ = δ( x −x ) δ( y −y ) δ( z −z ) dxdydz<br />
0 0 0 0 0 0<br />
∂(,, xyz)<br />
δ( r −r0) δ( θ −θ0) δ( φ − φ0) = δ( x −x0) δ( y −y0) δ( z −z0)<br />
∂(, r θφ , )<br />
<br />
Both sides vanish unless<br />
r = r +infinitesimal ≈ r ,<br />
0 0<br />
<strong>and</strong><br />
θ = θ +infinitesimal ≈ θ<br />
0 0 .<br />
Therefore, we can replace r with , <strong>and</strong> θ with θ .<br />
r0<br />
0<br />
r<br />
2<br />
sin θ<br />
2<br />
0 0 0 0 0 0 0 φ 0 )<br />
δ( x −x ) δ( y −y ) δ( z − z ) r sin θ = δ( r −r<br />
) δ( θ −θ ) δ(<br />
φ−<br />
. <br />
x = r sin θcosφ<br />
5.6 y = r sin θsin<br />
φ ,<br />
z = r cos θ<br />
r<br />
0<br />
= 0 ⇒<br />
1<br />
δ()()() x δ y δ z = δ()<br />
r<br />
2<br />
4πr<br />
Proof:<br />
δ()()() r δ θ δ φdrdθdφ = δ()()()<br />
x δ y δ z dxdydz<br />
=<br />
4<br />
1<br />
dr dr<br />
2 2<br />
χ 2 [ − , ]<br />
πr dr<br />
() r<br />
18
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
∂(,, xyz)<br />
δ()()() r δ θ δ φ = δ()()()<br />
x δ y δ z<br />
∂(, r θφ , )<br />
<br />
r<br />
2 sin θ<br />
Since r = , φ may take any value in [0,2 π]<br />
, θ may take any<br />
0<br />
0<br />
value in [0, π]<br />
, <strong>and</strong> we integrate over <strong>the</strong>m. Then,<br />
2<br />
φ= 2π φ=<br />
2π<br />
∫<br />
r sin θdθ dφδ( x)()() δ y δ z = δ()() r δ θ dθ δ( φ)<br />
dφ<br />
2<br />
φ= 0 φ=<br />
0<br />
<br />
2π<br />
1<br />
θ= π θ=<br />
π<br />
2 πr δ( x)()() δ y δ z sin θdθ = δ() r δ()<br />
θ dθ.<br />
2<br />
∫<br />
<br />
θ= 0 θ=<br />
0<br />
θ=<br />
π<br />
− cos θ = 2<br />
θ=<br />
0<br />
∫<br />
4 πr δ( x)()() δ y δ z = δ()<br />
r .<br />
<br />
1<br />
∫<br />
<br />
19
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
6.<br />
<strong>Radial</strong> <strong>Delta</strong> δ <strong>Radial</strong> () r<br />
6.1<br />
ν=∞<br />
sin(2 πνr)<br />
δ( x) δ( y) δ( z) = 4∫<br />
νdν<br />
r<br />
ν=<br />
0<br />
Proof: By 4.10,<br />
⎛ νx=∞ ⎞⎛<br />
νy<br />
=∞ ⎞⎛<br />
νz=∞<br />
2 i 2 i<br />
yy<br />
2<br />
( ) ( ) ( ) πνxx d i<br />
zz<br />
x<br />
πν<br />
d y<br />
πν<br />
δ δ δ = ν ν d ν<br />
⎜<br />
⎝νx=−∞ ⎠⎟⎜<br />
⎝<br />
νy=−∞ ⎠⎟<br />
⎝⎜νz=−∞<br />
Substitute<br />
x = r sin θcosφ<br />
νx<br />
= νsin<br />
γcos<br />
β<br />
y = r sin θsin<br />
φ νy<br />
= νsin<br />
γsin<br />
β<br />
z = r cos θ νz<br />
= νcos<br />
γ<br />
Then,<br />
ν x + ν y + ν z<br />
x y z<br />
=<br />
∫ ∫ ∫ z<br />
{ }<br />
= νr<br />
⎡sin θsin γ cosφcos β + sin φsin β + cos θcos<br />
γ⎤<br />
⎣<br />
⎦<br />
= νr<br />
⎡sin θsin γcos( β − φ) + cosθcos<br />
γ⎤<br />
⎣<br />
⎦<br />
Integrating with respect to ν , γ , <strong>and</strong> β ,<br />
ν=∞ ⎛γ= π⎡β=<br />
2π<br />
⎤ ⎞ 2πν i r sinθsinγcos( β−φ) 2πν i r cosθcosγ<br />
2<br />
= e dβ<br />
e<br />
sin γdγ ∫ ν dν<br />
∫ ∫<br />
ν= 0⎜<br />
⎝ ⎢<br />
⎥<br />
γ= 0⎣<br />
β=<br />
0<br />
⎦<br />
⎠⎟<br />
Denoting<br />
α = β − φ,<br />
⎞<br />
⎟<br />
⎠<br />
20
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
β= 2π α= 2π−φ<br />
2πν i r sin θsin γcos( β−φ) 2πν i r sin θsin γcos<br />
α<br />
∫ e dβ<br />
= ∫ e<br />
d α<br />
β= 0<br />
α=−φ<br />
2 i r sin sin cos<br />
Since e πν θ γ α is periodic in α with period 2 π ,<br />
α=<br />
2π<br />
α=<br />
0<br />
α=<br />
2π<br />
= ∫<br />
α=<br />
0<br />
e<br />
2πν i r sin θsin γcos<br />
α<br />
dα<br />
⎛<br />
2<br />
2πν i rsinθsinγcos α (2πν i rsinθsinγcos α)<br />
⎞<br />
= 1 .<br />
∫ ⎜<br />
+ + + .. dα<br />
.<br />
⎜⎝<br />
1 2!<br />
⎠⎟<br />
The integrals of <strong>the</strong> odd powers vanish, <strong>and</strong> we have<br />
2 4<br />
(2πνr sin θ sin γ) (2πνr sin θ sin γ) (2πνr<br />
sin θ sin γ)<br />
= 2π − 2π + 2π − 2π<br />
+ ...<br />
2 2 2 2 2 2<br />
2 2 ⋅4 2 ⋅4 ⋅6<br />
6<br />
=<br />
2 πJ<br />
(2πνr<br />
sinθsinγ).<br />
0<br />
Therefore,<br />
ν=∞ ⎛γ=<br />
π<br />
⎞ 2πν i r cosθcosγ<br />
2<br />
δ( x) δ( y) δ( z) = 2 πJ0(2πνrsin θsin γ) e sin γdγ ∫ ν dν<br />
∫<br />
ν= 0<br />
⎜<br />
⎝γ=<br />
0<br />
⎠⎟<br />
Denote<br />
Then,<br />
u = cos γ ,<br />
du =−sin<br />
γdγ<br />
γ = 0 ⇒ u = 1 ,<br />
γ = π ⇒ u = −1<br />
<strong>and</strong> <strong>the</strong> γ summation becomes<br />
21
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
u=<br />
1<br />
∫<br />
u=−1<br />
0<br />
2 [2πν<br />
i r cos θ]<br />
u<br />
2 π J ([2πνr sin θ] 1 − u ) e du =<br />
u=<br />
1<br />
∫<br />
= 8 π J ([2πνr sin θ] 1 −u )cos([2πνr cos θ] u)<br />
du.<br />
u=<br />
0<br />
0<br />
2<br />
Denote<br />
Then,<br />
2<br />
u = 1 − v .<br />
v<br />
du =− dv<br />
2<br />
1 − v<br />
u = 0 ⇒ v = 1 ,<br />
u = 1 ⇒ v = 0<br />
<strong>and</strong> <strong>the</strong> u summation becomes<br />
v=<br />
1<br />
∫<br />
= 8 π J ([2πνr sin θ] v)cos([2πνrcos θ] 1 −v<br />
)<br />
v=<br />
0<br />
0<br />
2<br />
v<br />
1 − v<br />
2<br />
dv.<br />
Denoting,<br />
b<br />
c<br />
=<br />
=<br />
2πνr<br />
cosθ<br />
,<br />
2πνr<br />
sinθ<br />
v=<br />
1<br />
2 v<br />
∫ J0<br />
cv b v dv.<br />
2<br />
v=<br />
0<br />
1 − v<br />
= 8 π ( )cos( 1 − )<br />
By [Prudnikov, Vol.2, p.201, 2.12.21 #5], <strong>the</strong> γ summation equals<br />
π<br />
= 8π<br />
+ +<br />
−1<br />
2 2<br />
( ) ( )<br />
4 2 2<br />
1<br />
2 b c J b c<br />
2<br />
22
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
Since 1 ( )<br />
π 1<br />
= 8π<br />
J1<br />
2 ν<br />
2 2πνr<br />
2<br />
1<br />
= 4π<br />
J1<br />
2 ν<br />
νr<br />
2<br />
( π r)<br />
( π r)<br />
2 J x = sin x , <strong>the</strong> γ summation equals<br />
2 πx<br />
= 1 2<br />
4π<br />
sin<br />
νr<br />
π2πνr<br />
2 ν<br />
( π r )<br />
Therefore,<br />
=<br />
1<br />
4 sin 2 r<br />
νr<br />
( πν )<br />
ν=∞<br />
ν=<br />
0<br />
( πνr<br />
)<br />
sin 2<br />
δ( x) δ( y) δ( z) = 4∫ νdν.<br />
<br />
r<br />
Thus,<br />
δ( x) δ( y) δ( z)<br />
is a <strong>Radial</strong>ly Symmetric <strong>Delta</strong> <strong>Function</strong> that we<br />
will denote δ r .<br />
<strong>Radial</strong> ()<br />
6.2 The <strong>Radial</strong> <strong>Delta</strong> r Definition<br />
<strong>Radial</strong> ()<br />
δ<br />
<strong>Radial</strong><br />
ν=∞<br />
sin ( 2πνr<br />
)<br />
δ<br />
() r ≡ 4∫ νdν<br />
r<br />
ν=<br />
0<br />
23
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
7.<br />
<strong>Radial</strong> <strong>Delta</strong><br />
δ <strong>Radial</strong> ( r − σ)<br />
By 6.2, we have,<br />
7.1<br />
δ<br />
<strong>Radial</strong><br />
( r −σ) ≡ 4<br />
ν=∞<br />
∫<br />
ν=<br />
0<br />
( πν r − σ )<br />
sin 2 [ ]<br />
r − σ<br />
νdν<br />
Alternatively, we may derive similarly to 6.1,<br />
ν=∞<br />
2 1<br />
7.2 δ( x −ξ) δ( y −η) δ( z − ζ) = 4 sin(2 πνr)sin(2 πνσ)<br />
ν<br />
rσ<br />
∫ d<br />
Proof: By 4.2,<br />
δ( x −ξ) δ( y −η) δ( z − ζ)<br />
=<br />
ν=<br />
0<br />
ν=∞<br />
2 1<br />
=<br />
2 ∫<br />
r<br />
ν=<br />
0<br />
4 sin(2 πνr)sin(2 πνσ)<br />
dν<br />
⎛ νx<br />
=∞ ⎞⎛<br />
νy<br />
=∞<br />
⎞⎛<br />
νz<br />
=∞<br />
e 2 πν i ( ) 2 i<br />
y<br />
( y )<br />
x<br />
x−ξ d 2 i<br />
z<br />
( z )<br />
x<br />
e πν −η<br />
d y<br />
e πν −ζ<br />
= ν<br />
ν<br />
∫ ∫ ∫ d<br />
⎜⎝ ν =−∞ ⎟⎠⎜<br />
⎝ν =−∞ ⎠⎟<br />
⎜⎝ν<br />
=−∞<br />
x y z<br />
ν z<br />
⎞<br />
⎠⎟<br />
Substitute<br />
x = r sin θcosφ<br />
y = r sin θsin<br />
φ<br />
z = r cos θ<br />
Then,<br />
ξ = σsin<br />
λcos<br />
μ<br />
η = σsin<br />
λsin<br />
μ<br />
ζ = σcosλ<br />
ν = νsin<br />
γcos<br />
β<br />
x<br />
ν = νsin<br />
γsin<br />
β<br />
y<br />
ν = νcos<br />
γ<br />
z<br />
24
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
ν x + ν y + ν z =<br />
x y z<br />
νξ<br />
x<br />
+ νη<br />
y<br />
+ νζ<br />
z<br />
= νr<br />
⎡sin θsin γ cosφcos β + sin φsin β + cos θcos<br />
γ⎤<br />
⎣<br />
⎦<br />
= νr<br />
⎡sin θsin γcos( β − φ) + cosθcos<br />
γ⎤<br />
⎣<br />
⎦<br />
=<br />
{ }<br />
{ }<br />
= νσ ⎡sinλ sin γ cos μ cos β + sin μ sin β + cosλ cos γ ⎤<br />
⎣<br />
⎦<br />
= νσ ⎡sin λ sin γ cos( β − μ) + cosλ cos γ ⎤<br />
⎣<br />
⎦<br />
Integrating with respect to ν , γ , <strong>and</strong> β ,<br />
ν=∞ ⎛γ= π⎛β=<br />
2π<br />
⎞ 2πν i r sin θsin γcos( β−φ) −2πνσ i sin λsin γcos( β−μ)<br />
= e<br />
e<br />
dβ<br />
∫ ×<br />
∫ ∫<br />
ν= 0 ⎜ ⎝γ= 0⎜⎝ β=<br />
0<br />
⎠⎟<br />
2πν i r cosθcos γ −2πνσ i cosλcos γ<br />
2<br />
× e e sin γdγ ν<br />
)<br />
dν.<br />
Denoting<br />
A = β − φ, B = β −μ,<br />
β=<br />
2π<br />
∫<br />
β=<br />
0<br />
e<br />
2πν i r sin θsin γcos( β−φ) −2πνσ i sinλsin γcos( β−μ)<br />
e<br />
dβ<br />
=<br />
A= 2π− φ B= 2π−μ<br />
2πν i r sin θsin γcosA −2πνσ i sinλsin γcosB<br />
= ∫ ∫<br />
e dA e<br />
A=− φ<br />
B=−μ<br />
dB<br />
2πν i rsinθsinγcosA<br />
−2πνσ i sinλsinγcosB<br />
e<br />
Since e , <strong>and</strong> are periodic with<br />
period 2 π ,<br />
25
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
A= 2π<br />
B=<br />
2π<br />
2πν i r sin θsin γcosA −2πνσ i sin λsin γcosB<br />
= ∫ e dA ∫ e<br />
dB<br />
A= 0 B=<br />
0<br />
Integrating,<br />
A=<br />
2π<br />
∫<br />
A=<br />
0<br />
e<br />
2πν i r sinθsinγcosA<br />
dA<br />
=<br />
A=<br />
2π<br />
⎛<br />
2<br />
2πν i rsinθsinγcos A (2πν i rsinθsinγcos A)<br />
⎞<br />
= 1 ...<br />
∫<br />
⎜<br />
+ + +<br />
dA<br />
⎜⎝<br />
1 2!<br />
⎠⎟<br />
A=<br />
0<br />
The integrals of <strong>the</strong> odd powers vanish, <strong>and</strong> we have<br />
2 4<br />
(2πνr sin θ sin γ) (2πνr sin θ sin γ) (2πνr<br />
sin θ sin γ)<br />
= 2π − 2π + 2π − 2π<br />
+ ...<br />
2 2 2 2 2 2<br />
2 2 ⋅4 2 ⋅4 ⋅6<br />
6<br />
=<br />
2 πJ<br />
(2πνrsinθsinγ).<br />
0<br />
Similarly,<br />
B=<br />
2π<br />
∫<br />
B=<br />
0<br />
e<br />
−2πνσ i sinλsinγcosB<br />
dB<br />
=<br />
B=<br />
2π<br />
⎛<br />
2<br />
2πνσ i sinλsin γcos B (2πν i rsinλsin γcos B)<br />
⎞<br />
= 1 .<br />
∫ ⎜<br />
+ + + ..<br />
dB<br />
⎜⎝<br />
1 2!<br />
⎠⎟<br />
B=<br />
0<br />
The integrals of <strong>the</strong> odd powers vanish, <strong>and</strong> we have<br />
2 4<br />
(2πνσsin λsin γ) (2πνσsin λsin γ) (2πνσsin λsin γ)<br />
= 2π − 2π + 2π − 2π<br />
+ ...<br />
2 2 2 2 2 2<br />
2 2 ⋅4 2 ⋅4 ⋅6<br />
6<br />
=<br />
2 πJ<br />
(2πνσsinλsinγ).<br />
0<br />
26
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
Therefore, <strong>the</strong><br />
γ<br />
summation is<br />
γ=<br />
π<br />
∫<br />
γ=<br />
0<br />
2 πJ (2πνr sinθsin γ)<br />
e<br />
0<br />
2πν i r cosθcosγ<br />
0<br />
×<br />
−2 i cos cos<br />
πνσ λ γ<br />
× 2 πJ (2πνσsinλsin γ) e sin γdγ<br />
=<br />
Denoting,<br />
b<br />
c<br />
1<br />
1<br />
= 2πνrcosθ<br />
b2<br />
= 2πνσ<br />
cosλ<br />
, ,<br />
= 2πνr<br />
sinθ<br />
c = 2πνσ<br />
sinλ<br />
2<br />
γ=<br />
π<br />
= ∫<br />
γ=<br />
0<br />
b cos γ<br />
0 1 0 2<br />
−b<br />
cos γ<br />
2 πJ ( c sin γ) e<br />
1<br />
2 πJ ( c sin γ) e<br />
2<br />
sin γdγ<br />
Denote<br />
Then,<br />
u = cos γ ,<br />
du =−sin<br />
γdγ<br />
γ = 0 ⇒ u = 1 ,<br />
γ = π ⇒ u = −1<br />
<strong>and</strong> <strong>the</strong> γ summation becomes<br />
u=<br />
1<br />
∫<br />
u=−1<br />
2 bu<br />
2<br />
0 1 0 2<br />
= 2 π ( 1 − ) 2 ( 1 − )<br />
−bu<br />
J c u e<br />
1<br />
πJ c u e<br />
2<br />
du<br />
u= 1 u=<br />
1<br />
∫<br />
2 bu<br />
π<br />
0 1 ∫ 0 2<br />
2<br />
u=− 1 u=−1<br />
= 2 ( 1 − ) 2 ( 1 − )<br />
−bu<br />
J c u e<br />
1<br />
du πJ c u e<br />
2<br />
du<br />
The first integration is<br />
27
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
u= 1 u=<br />
1<br />
∫<br />
2 bu<br />
2<br />
0 1 ∫ 0 1 1<br />
u=− 1 u=<br />
0<br />
2 π ( 1 − )<br />
1<br />
= 8 π ( 1 − )cos( )<br />
J c u e du J c u bu du<br />
Denote<br />
Then,<br />
2<br />
u = 1 − v .<br />
v<br />
du =− dv<br />
2<br />
1 − v<br />
u = 0 ⇒ v = 1 ,<br />
u = 1 ⇒ v = 0<br />
<strong>and</strong> <strong>the</strong> u summation becomes<br />
v=<br />
1<br />
2 v<br />
∫ J0 c1v b1<br />
v dv.<br />
2<br />
v=<br />
0<br />
1 − v<br />
= 8 π ( )cos( 1 − )<br />
By [Prudnikov, Vol.2, p.201, 2.12.21 #5], <strong>the</strong> v summation equals<br />
b1<br />
= 2πνrcosθ<br />
Recalling ,<br />
c = 2πνrsinθ<br />
1<br />
Since 1 ( )<br />
π<br />
= 8π<br />
+ +<br />
−1<br />
2 2<br />
( ) ( )<br />
4 2 2<br />
1 1 1 1 1<br />
2 b c J b c<br />
2<br />
π 1<br />
= 8π<br />
J1<br />
2 ν<br />
2 2πνr<br />
2<br />
1<br />
= 4π<br />
J1<br />
2 ν<br />
νr<br />
2<br />
( π r)<br />
( π r)<br />
2 J x = sin x , <strong>the</strong> v summation equals<br />
2 πx<br />
28
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
= 1 2<br />
4π<br />
sin<br />
νr<br />
π2πνr<br />
2 ν<br />
( π r )<br />
That is,<br />
1<br />
= 4 sin( 2πνr<br />
).<br />
νr<br />
u=<br />
1<br />
( π )<br />
2 bu<br />
1<br />
∫ 2 π<br />
0( 1<br />
1 − ) = 4 sin 2 ν<br />
u=−1<br />
1<br />
J c u e du r<br />
νr<br />
Similarly,<br />
u=<br />
1<br />
∫<br />
u=−1<br />
2<br />
2 ( 1 )<br />
2<br />
4 sin 2<br />
0 2<br />
bu<br />
πJ c − u e du =<br />
1<br />
νσ<br />
( πνσ<br />
)<br />
Therefore, <strong>the</strong><br />
γ<br />
summation is<br />
<strong>and</strong> <strong>the</strong> ν summation is<br />
4 1 sin 2 4 1 sin 2<br />
νr<br />
νσ<br />
( πνr<br />
) ( πνσ )<br />
ν=∞ ν=∞<br />
1 1 2 2 1<br />
4 sin( 2πνr) 4 sin( 2πνσ ) ν dν = 4 sin( 2πνσ ) sin( 2πν ) ν<br />
νr<br />
νσ rσ<br />
∫ ∫ r d<br />
ν= 0 ν=<br />
0<br />
Therefore,<br />
ν=∞<br />
2 1<br />
δ( x −ξ) δ( y −η) δ( z − ζ) = 4 sin(2 πνr)sin(2 πνσ)<br />
dν<br />
rσ<br />
∫ . <br />
ν=<br />
0<br />
Thus, δ( x −ξ) δ( y −η) δ(<br />
z −ζ)<br />
is a <strong>Radial</strong>ly Symmetric <strong>Delta</strong><br />
function that we will denote<br />
δ <strong>Radial</strong> ( r − σ)<br />
29
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
7.3 The <strong>Radial</strong> <strong>Delta</strong> δ − ) Definition<br />
<strong>Radial</strong> (r<br />
ν=∞<br />
ν=<br />
0<br />
σ ( ) ( r)<br />
2 1<br />
δ<strong>Radial</strong>( r −σ) ≡ 4 sin 2πνσ sin 2πν<br />
rσ<br />
∫ dν<br />
spikes at r = σ , <strong>and</strong> vanishes for r ≠ σ .<br />
From 7.1, <strong>and</strong> 7.3,<br />
7.4<br />
ν<br />
( πν r − σ ) ( πνσ ) ( πνr)<br />
ν=∞ =∞<br />
sin 2 [ ] sin 2 sin 2<br />
νdν<br />
= 4<br />
r − σ<br />
rσ<br />
∫ ∫<br />
ν= 0 ν=<br />
0<br />
dν<br />
Common Tables do not have 7.4.<br />
30
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
8.<br />
δ ( r − r ) <strong>and</strong><br />
<strong>Radial</strong> 0<br />
δ( r − r )<br />
0<br />
δ<br />
1<br />
( r − r ) = δ( r − r )<br />
8.1<br />
<strong>Radial</strong> 0 2 0<br />
r0<br />
x = r sin θcosφ<br />
Proof: Denote y = rsin<br />
θsin<br />
φ , y = r sin θ sin φ ,<br />
z = rcos<br />
θ<br />
Then,<br />
x<br />
z<br />
= r<br />
sin θ<br />
0 0 0<br />
0 0 0<br />
= r<br />
cosθ<br />
0 0 0<br />
cosφ<br />
δ( x −x ) δ( y −y ) δ( z − z ) dxdydz = δ( r −r )( dr)(sin θd θ)<br />
dφ<br />
0 0 0 0<br />
∂(,, xyz)<br />
δ( x −x0) δ( y −y0) δ( z − z0) = δ(<br />
r −r<br />
∂(, r −cos θφ , )<br />
0<br />
0<br />
0 )<br />
Since<br />
∂(,, xyz)<br />
∂(, r −cos θφ , )<br />
=<br />
r<br />
2<br />
,<br />
δ( x −x0) δ( y −y0) δ( z − z0) = δ(<br />
r −r<br />
2 0<br />
r<br />
)<br />
δ ( r−r<br />
)<br />
δ<br />
<strong>Radial</strong> 0<br />
1<br />
( r − r ) = δ( r − r ).<br />
<br />
<strong>Radial</strong> 0 2 0<br />
r0<br />
1<br />
8.2 Sifting by δ <strong>Radial</strong> ( r σ)<br />
− 31
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
σ=∞<br />
2<br />
∫ δ<strong>Radial</strong>( r − σ)<br />
σ dσ<br />
= 1<br />
σ=<br />
0<br />
σ=∞ σ=∞<br />
∫<br />
Proof: By 8.1, δ ( r − σ) σ 2 dσ = δ( r −σ) dσ<br />
= 1.<br />
∫<br />
<strong>Radial</strong><br />
σ= 0 σ=<br />
0<br />
ν=∞<br />
2<br />
8.3 δ( r − σ) = 4 ∫ sin(2 πνr)sin(2 πνσ)<br />
dν<br />
ν=<br />
0<br />
Proof: By 8.1,<br />
=<br />
4<br />
ν=∞<br />
∫<br />
ν=<br />
0<br />
( r − )<br />
sin 2 πν[ σ]<br />
ν d ν<br />
r − σ<br />
2<br />
δ( r − σ) = σ δ ( r −σ)<br />
<strong>Radial</strong><br />
ν=∞<br />
2 2<br />
= σ 4 ∫<br />
ν=<br />
0<br />
sin(2 πνr)sin(2 πνσ)<br />
dν<br />
rσ<br />
For r ≠ σ , <strong>the</strong> integral vanishes. For r = σ , we have<br />
ν=∞<br />
2 2<br />
= σ 4 ∫<br />
By 7.3,<br />
= 4<br />
ν=<br />
0<br />
ν=∞<br />
2<br />
ν=<br />
0<br />
sin(2 πνr)sin(2 πνσ)<br />
dν<br />
2<br />
σ<br />
= 4 ∫ sin(2 πνr)sin(2 πνσ)<br />
dν<br />
ν=∞<br />
ν=<br />
0<br />
Common Tables do not have 8.3.<br />
( r − )<br />
sin 2 πν[ σ]<br />
∫ ν d ν .<br />
r − σ<br />
32
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
9.<br />
3-D <strong>Fourier</strong>-<strong>Bessel</strong> Transform<br />
The 3-dimensional <strong>Fourier</strong>-<strong>Bessel</strong> Transform is <strong>the</strong> 3-dimensional<br />
<strong>Fourier</strong> Transform applied to a <strong>Radial</strong>ly symmetric function f () r .<br />
Then, integration of <strong>the</strong> Exponential <strong>Function</strong> with respect to <strong>the</strong><br />
azimuth angle, yields a <strong>Bessel</strong> <strong>Function</strong>.<br />
9.1 The 3-dimensional <strong>Fourier</strong>-<strong>Bessel</strong> Transform<br />
F<br />
Bess<br />
sin 2<br />
() = 4 ∫ f()<br />
r<br />
ν<br />
{ f r }<br />
r =∞<br />
r = 0<br />
( πνr<br />
)<br />
rdr<br />
Proof:<br />
The 3-dimensional <strong>Fourier</strong> Transform of a radially<br />
symmetric function f (,, xyz) = fr () is<br />
F<br />
Bess<br />
{ }<br />
z=∞ y=∞ x=∞<br />
−2πν i x−2πν i y−2πν<br />
i z<br />
x y z<br />
f () r f () r e dxdydz<br />
= ∫ ∫ ∫<br />
z=−∞ y=−∞ x=−∞<br />
Substitute<br />
Then,<br />
x<br />
y<br />
z<br />
= r sin θcosφ<br />
= r sin θsin<br />
φ<br />
= r cosθ<br />
ν = νsin<br />
γcos<br />
β<br />
x<br />
ν = νsin<br />
γsin<br />
β<br />
y<br />
ν = νcos<br />
γ<br />
z<br />
33
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
ν x + ν y + ν z =<br />
x y z<br />
{ }<br />
= νr<br />
⎡sin θsin γ cosφcos β + sin φsin β + cos θcos<br />
γ⎤<br />
⎣<br />
⎦<br />
= νr<br />
⎡sin θsin γcos( β − φ) + cosθcos<br />
γ⎤<br />
⎣<br />
⎦<br />
Integrating with respect to r , θ , <strong>and</strong> φ ,<br />
F<br />
Bess<br />
r =∞ ⎛θ= π⎡φ=<br />
2π<br />
⎤ ⎞ −2πν i r sinθsinγcos( φ−β) −2πν i r cosθcosγ<br />
2<br />
f () r = f() r ∫ e dφ<br />
e sinθd<br />
∫ ∫<br />
θ<br />
r dr<br />
⎢<br />
⎥<br />
r = 0<br />
⎜⎝θ= 0 φ=<br />
0<br />
⎠⎟<br />
⎣<br />
⎦<br />
{ }<br />
Denoting<br />
α<br />
= φ− β,<br />
φ= 2π α= 2π−β<br />
−2πν i r sinθsinγcos( φ−β) −2πν i r sinθsinγcosα<br />
∫ ∫<br />
e dφ<br />
= e<br />
φ= 0<br />
α=−β<br />
dα<br />
−2πν i r sinθsinγcosα<br />
Since e<br />
is periodic in α with period 2 π ,<br />
α=<br />
2π<br />
α=<br />
0<br />
α=<br />
2π<br />
= ∫<br />
α=<br />
0<br />
e<br />
−2πν i r sin θsin γcos<br />
α<br />
dα<br />
⎛<br />
2<br />
2πν i rinθsinγcos α (2πν i rsinθsinγcos α)<br />
⎞<br />
= 1 ...<br />
∫ ⎜<br />
− + −<br />
dα<br />
.<br />
⎜⎝<br />
1 2!<br />
⎠⎟<br />
The integrals of <strong>the</strong> odd powers vanish, <strong>and</strong> we have<br />
2 4<br />
(2πνr sin θ sin γ) (2πνr sin θ sin γ) (2πνr<br />
sin θ sin γ)<br />
= 2π − 2π + 2π − 2π<br />
+ ...<br />
2 2 2 2 2 2<br />
2 2 ⋅4 2 ⋅4 ⋅6<br />
6<br />
=<br />
2 πJ<br />
(2πνrsinθsinγ).<br />
0<br />
Therefore,<br />
34
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
F<br />
Bess<br />
⎛<br />
r =∞ θ=<br />
π<br />
⎜<br />
= ∫<br />
∫ 0<br />
r = 0<br />
⎜<br />
θ=<br />
0<br />
⎟<br />
⎞ ⎟<br />
r dr<br />
⎠<br />
2 i r cos cos 2<br />
{ f () r } f()2 r ⎜<br />
− πν θ γ<br />
π J (2πνrsinθsin γ) e sinθdθ⎟<br />
⎝<br />
Denoting,<br />
b<br />
c<br />
=<br />
=<br />
2πνr<br />
cosγ<br />
,<br />
2πνrsinγ<br />
<strong>the</strong><br />
θ<br />
summation becomes<br />
θ=<br />
π<br />
−b<br />
cos θ<br />
2 π∫ J0( csin θ) e sinθdθ.<br />
θ=<br />
0<br />
Denote<br />
Then,<br />
u = cos θ .<br />
du =−sin<br />
θdθ<br />
θ = 0 ⇒ u = 1 ,<br />
θ = π ⇒ u = −1<br />
<strong>and</strong> <strong>the</strong> θ summation becomes<br />
u= 1 u=<br />
1<br />
∫<br />
2 bu<br />
0<br />
− = π∫<br />
0<br />
−<br />
2<br />
u=− 1 u=<br />
0<br />
2 π J ( c 1 u ) e du 8 J ( c 1 u )cos( bu)<br />
du<br />
Denote<br />
Then,<br />
u<br />
2<br />
= 1 − v .<br />
35
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
v<br />
du =− dv<br />
2<br />
1 − v<br />
u = 0 ⇒ v = 1 ,<br />
u = 1 ⇒ v = 0<br />
<strong>and</strong> <strong>the</strong> u summation becomes<br />
v=<br />
1<br />
2 v<br />
= 8 π ∫ J0( cv)cos( b 1 −v ) dv<br />
2<br />
1 − v<br />
v=<br />
0<br />
By [Prudnikov, Vol.2, p.201, 2.12.21 #5], this equals<br />
b = 2πνr<br />
cosγ<br />
Recalling ,<br />
c = 2πνrsinγ<br />
Since 1 ( )<br />
π<br />
= 8π<br />
+ +<br />
−1<br />
2 2<br />
( ) ( )<br />
4 2 2<br />
1<br />
2 b c J b c<br />
2<br />
π 1<br />
= 8π<br />
J1<br />
2 ν<br />
2 2πνr<br />
2<br />
1<br />
= 4π<br />
J1<br />
2 ν<br />
νr<br />
2<br />
( π r)<br />
( π r)<br />
2 J x = sin x , <strong>the</strong> v summation equals<br />
2 πx<br />
= 1 2<br />
4π<br />
sin<br />
νr<br />
π2πνr<br />
2 ν<br />
( π r )<br />
Therefore,<br />
=<br />
1<br />
4 sin 2 r<br />
νr<br />
( πν )<br />
36
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
F<br />
Bess<br />
r =∞<br />
{ f () r } f()4 r sin( 2πνr) 2<br />
= ∫<br />
r = 0<br />
r =∞<br />
r = 0<br />
1<br />
νr<br />
( πνr<br />
)<br />
r dr<br />
sin 2<br />
= 4 ∫ f ( r)<br />
rdr .<br />
ν<br />
9.2 3-D Inverse <strong>Fourier</strong>-<strong>Bessel</strong> Transform<br />
F<br />
−1<br />
Bess<br />
ν=∞<br />
ν=<br />
0<br />
( πνr<br />
)<br />
sin 2<br />
F(2 πν) = 4 ∫ F(2 πν)<br />
νdν<br />
r<br />
Proof: The 3-dimensional Inverse <strong>Fourier</strong>-<strong>Bessel</strong> Transform of a<br />
radially symmetric function<br />
F( ν , ν , ν ) = F(2 πν)<br />
x y z<br />
is<br />
F<br />
νz=∞ νy<br />
=∞ νx=∞<br />
−1<br />
2 πi[ νxx+ νyy+<br />
νzz]<br />
Bess { F(2 πν) } = ∫ ∫ ∫ F(2 πν)<br />
e<br />
dνxdνydνz<br />
ν =−∞ ν =−∞ ν =−∞<br />
Substitute<br />
z y x<br />
Then,<br />
x<br />
y<br />
z<br />
= r sin θcosφ<br />
= r sin θsin<br />
φ<br />
= r cosθ<br />
ν = νsin<br />
γcos<br />
β<br />
x<br />
ν = νsin<br />
γsin<br />
β<br />
y<br />
ν = νcos<br />
γ<br />
z<br />
ν x + ν y + ν z<br />
x y z<br />
=<br />
{ }<br />
= νr<br />
⎡sin θsin γ cosφcos β + sin φsin β + cos θcos<br />
γ⎤<br />
⎣<br />
⎦<br />
= νr<br />
⎡sin θsin γcos( β − φ) + cosθcos<br />
γ⎤<br />
⎣<br />
⎦<br />
Integrating with respect to ν , γ , <strong>and</strong> β ,<br />
37
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
ν=∞ ⎛γ= π⎡β=<br />
2π<br />
⎤ ⎞<br />
1 2 i r sin sin cos( ) 2 i r cos cos 2<br />
Bess { F(2 )}<br />
−<br />
πν θ γ β−φ πν θ γ<br />
F πν = F(2 πν) ∫ e dβ e sin γdγ ν ν<br />
∫ ∫<br />
d<br />
ν= 0<br />
⎜ ⎢<br />
⎥<br />
⎝γ= 0 β=<br />
0<br />
⎠⎟<br />
⎣<br />
⎦<br />
Denoting<br />
α<br />
= β −φ,<br />
β= 2π α= 2π−φ<br />
2πν i r sin θsin γcos( β−φ) 2πν i r sin θsin γcos<br />
α<br />
∫ ∫<br />
e dβ<br />
= e<br />
β= 0<br />
α=−φ<br />
dα<br />
2 i r sin sin cos<br />
Since e πν θ γ α is periodic in α with period 2 π ,<br />
α=<br />
2π<br />
α=<br />
0<br />
α=<br />
2π<br />
= ∫<br />
α=<br />
0<br />
e<br />
2πν i r sin θsin γcos<br />
α<br />
dα<br />
⎛<br />
2<br />
2πν i rsinθsinγcos α (2πν i rsinθsinγcos α)<br />
⎞<br />
= 1 .<br />
∫ ⎜<br />
+ + + .. dα<br />
.<br />
⎜⎝<br />
1 2!<br />
⎠⎟<br />
The integrals of <strong>the</strong> odd powers vanish, <strong>and</strong> we have<br />
2 4<br />
(2πνr sin θ sin γ) (2πνr sin θ sin γ) (2πνr<br />
sin θ sin γ)<br />
= 2π − 2π + 2π − 2π<br />
+ ...<br />
2 2 2 2 2 2<br />
2 2 ⋅4 2 ⋅4 ⋅6<br />
6<br />
=<br />
2 πJ<br />
(2πνrsinθsinγ).<br />
0<br />
Therefore,<br />
ν=∞ ⎛γ=<br />
π<br />
⎞<br />
−1 2 i r cos cos<br />
(2 ) (2 ) 2 2 Bess F F J 0(2 r sin sin ) e πν θ γ<br />
F πν = πν π πν θ γ sin γ d γ ∫<br />
ν dν<br />
∫<br />
ν= 0<br />
⎜⎝γ=<br />
0<br />
⎠⎟<br />
Denoting,<br />
b<br />
c<br />
=<br />
=<br />
2πνr<br />
cosθ<br />
,<br />
2πνr<br />
sinθ<br />
<strong>the</strong><br />
γ<br />
summation is<br />
38
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
Denote<br />
Then,<br />
γ=<br />
π<br />
b cos γ<br />
∫ 2 πJ0( csin γ) e sin γdγ.<br />
γ=<br />
0<br />
u = cos γ<br />
du =−sin<br />
γdγ<br />
γ = 0 ⇒ u = 1 ,<br />
γ = π ⇒ u = −1<br />
<strong>and</strong> <strong>the</strong> γ summation becomes<br />
u= 1 u=<br />
1<br />
∫<br />
2 bu<br />
0<br />
− = π∫<br />
0<br />
−<br />
2<br />
u=− 1 u=<br />
0<br />
2 π J ( c 1 u ) e du 8 J ( c 1 u )cos( bu)<br />
du<br />
Denote<br />
Then,<br />
2<br />
u = 1 − v .<br />
v<br />
du =− dv<br />
2<br />
1 − v<br />
u = 0 ⇒ v = 1 ,<br />
u = 1 ⇒ v = 0<br />
<strong>and</strong> <strong>the</strong> u summation becomes<br />
v=<br />
1<br />
2 v<br />
= 8 π ∫ J0( cv)cos( b 1 −v ) dv<br />
2<br />
1 − v<br />
v=<br />
0<br />
By [Prudnikov, Vol.2, p.201, 2.12.21 #5], this equals<br />
π<br />
= 8π<br />
+ +<br />
−1<br />
2 2<br />
( ) ( )<br />
4 2 2<br />
1<br />
2 b c J b c<br />
2<br />
39
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
Since 1 ( )<br />
π 1<br />
= 8π<br />
J1<br />
2 ν<br />
2 2πνr<br />
2<br />
1<br />
= 4π<br />
J1<br />
2 ν<br />
νr<br />
2<br />
2 J x = sin x ,<br />
2 πx<br />
( π r)<br />
( π r)<br />
= 1 2<br />
4π<br />
sin<br />
νr<br />
π2πνr<br />
2 ν<br />
( π r )<br />
Therefore,<br />
=<br />
1<br />
4 sin 2 r<br />
νr<br />
( πν )<br />
ν=∞<br />
ν=<br />
0<br />
( πνr<br />
)<br />
sin 2<br />
−1<br />
FBessF(2 πν) = 4 ∫ F(2 πν)<br />
νd ν .<br />
r<br />
40
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
10.<br />
<strong>Fourier</strong>-<strong>Bessel</strong> Integral Theorem<br />
Holds only in Infinitesimal Calculus<br />
The 3-dimensional <strong>Fourier</strong>-<strong>Bessel</strong> Integral Theorem guarantees<br />
that <strong>the</strong> 3-dimensional <strong>Fourier</strong>-<strong>Bessel</strong> Transform <strong>and</strong> its Inverse<br />
are well defined operations, so that inversion yields <strong>the</strong> original<br />
function that generated <strong>the</strong> Transform.<br />
That is,<br />
ν=∞ ⎛ σ=∞<br />
sin 2<br />
⎞ sin 2<br />
f () r = 4 4 f()<br />
σ<br />
σdσ<br />
ν<br />
r<br />
ν= 0⎜<br />
⎝ σ=<br />
0<br />
⎠⎟<br />
( πνσ ) ( πνr<br />
)<br />
∫ ∫ d<br />
But in <strong>the</strong> Calculus of Limits, this integral is singular, <strong>and</strong> <strong>the</strong><br />
<strong>Fourier</strong>-<strong>Bessel</strong> Integral Theorem does not hold in <strong>the</strong> Calculus of<br />
Limits under any conditions.<br />
ν ν<br />
10.1 The 3-D <strong>Fourier</strong>-<strong>Bessel</strong> Integral Theorem Fails<br />
in <strong>the</strong> Calculus of Limits<br />
Proof: By <strong>the</strong> 3-dimensional <strong>Fourier</strong>-<strong>Bessel</strong> Integral Theorem<br />
41
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
⎛<br />
⎞ ⎟<br />
f () r = 4 ⎜4 f()<br />
d ⎟<br />
⎝<br />
⎠<br />
( πνσ ) ( πνr<br />
)<br />
ν=∞ σ=∞<br />
sin 2 sin 2<br />
⎜<br />
2 2<br />
σ σ σ<br />
∫ ∫ ν d<br />
νσ<br />
νr<br />
ν= 0 ⎜ σ=<br />
0<br />
⎟<br />
σ=∞ ⎛ ν=∞<br />
⎞ 2 1<br />
2<br />
= f ( σ) 4 sin( 2πνσ) sin( 2πνr)<br />
d d<br />
rσ<br />
σ= 0 ⎜<br />
⎝ ν=<br />
0<br />
⎠⎟<br />
However, at σ = r ,<br />
∫ ∫ ν σ σ<br />
ν=∞ ν=∞<br />
∫<br />
2<br />
( ) ( r) d = ( )<br />
sin 2πνσ sin 2πν ν sin 2πνσ dν<br />
ν= 0 ν=<br />
0<br />
∫<br />
ν=∞<br />
1<br />
= −<br />
2<br />
∫<br />
ν=<br />
0<br />
{ 1 cos( 4πνσ<br />
)}<br />
ν =∞<br />
⎡ sin(4 πνσ)<br />
⎤<br />
= ν − = ∞.<br />
⎣<br />
⎢ 4πσ<br />
⎦<br />
⎥<br />
ν = 0<br />
dν<br />
ν<br />
Hence, <strong>the</strong> 3-dimensinal <strong>Fourier</strong>-<strong>Bessel</strong> Integral diverges, <strong>and</strong> <strong>the</strong><br />
3-dimensional <strong>Fourier</strong>-<strong>Bessel</strong> Integral Theorem does not hold in<br />
<strong>the</strong> Calculus of Limits.<br />
Avoiding <strong>the</strong> singularity at σ = ρ does not recover <strong>the</strong> Theorem,<br />
because without <strong>the</strong> singularity <strong>the</strong> integral equals zero.<br />
Fur<strong>the</strong>rmore,<br />
10.2 Calculus of Limits Conditions are insufficient<br />
for <strong>the</strong> 3-D <strong>Fourier</strong>-<strong>Bessel</strong> Integral Theorem<br />
42
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
Proof: According to [Watson, p.458], The following conditions<br />
guarantee <strong>the</strong> 3-dimensional <strong>Fourier</strong>-<strong>Bessel</strong> Integral Theorem in<br />
<strong>the</strong> Calculus of Limits<br />
r =∞<br />
1. convergence of f () r<br />
∫<br />
r =−∞<br />
rdr<br />
r =∞ ⎛ω=<br />
λ<br />
⎞ 2. Existence of lim f ( r) J1( ωσ) J1( ωr)<br />
ωdω<br />
r<br />
λ→∞ 2 2<br />
r = 0 ⎜<br />
⎝ω<br />
= 0<br />
⎠⎟<br />
∫ ∫ dr<br />
It is clear from 7.1 that Condition 2. never holds. No conditions<br />
on f ( x ) can resolve <strong>the</strong> singularity, at<br />
ω=∞ ν=∞<br />
2<br />
1 1<br />
2 2<br />
ω= 0 ν=<br />
0<br />
σ = r , of<br />
∫ ∫ .<br />
r J ( ωσ) J ( ωr) ωdω = 4 sin(2 πνσ)sin(2 πνr)<br />
dν<br />
Therefore, <strong>the</strong> Calculus of Limits Conditions are insufficient for<br />
<strong>the</strong> 3-dimensional <strong>Fourier</strong> Integral Theorem. <br />
In Infinitesimal Calculus, <strong>the</strong> 3-dimensional <strong>Fourier</strong>-<strong>Bessel</strong><br />
Integral Theorem holds for any radial Hyper-real function<br />
10.3 3-dimensional <strong>Fourier</strong> Integral Theorem<br />
If f (,, xyz) = fr () is Hyper-real function,<br />
Then, <strong>the</strong> <strong>Fourier</strong>-<strong>Bessel</strong> Integral Theorem holds.<br />
⎛<br />
⎞ ⎟<br />
f () r = 4 ⎜4 f()<br />
d ⎟<br />
⎝<br />
⎠<br />
( πνσ ) ( πνr<br />
)<br />
ν=∞ σ=∞<br />
sin 2 sin 2<br />
⎜<br />
2 2<br />
σ σ σ<br />
∫ ∫ ν d<br />
νσ<br />
νr<br />
ν= 0<br />
⎜<br />
σ=<br />
0<br />
⎟<br />
ν<br />
43
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
σ=∞ ⎛ ν=∞<br />
⎞ 2 1<br />
2<br />
= f ( σ) 4 sin( 2πνσ) sin( 2πνr)<br />
d d<br />
rσ<br />
σ= 0<br />
⎜<br />
⎝ ν=<br />
0<br />
⎠⎟<br />
∫ ∫ ν σ σ.<br />
Proof:<br />
f () r = f(,, x y z )<br />
In Infinitesimal Calculus,<br />
By 7.2,<br />
ζ=∞ η=∞ ξ=∞<br />
∫ ∫ ∫ d<br />
= f ( ξηζδ , , ) ( x −ξδ ) ( y −ηδ ) ( z − ζ)<br />
dξdηζ<br />
ζ=−∞ η=−∞ ξ=−∞<br />
ν=∞<br />
2 1<br />
δ( x −ξ, y −η, z − ζ) = 4 sin( 2πνσ) sin( 2πνr)<br />
d<br />
rσ<br />
∫ ν,<br />
ν=<br />
0<br />
<strong>and</strong><br />
ζ=∞ η=∞ ξ=∞ ⎛ ν=∞<br />
⎞ 2 1<br />
f ( r) = f( ξηζ , , ) 4 sin( 2πνσ) sin( 2πνr)<br />
dν ∫ ∫ ∫ ξ ηζ<br />
rσ<br />
∫ d d d<br />
ζ=−∞ η=−∞ ξ=−∞ ⎜⎝<br />
ν=<br />
0<br />
⎠⎟<br />
Put<br />
f ( ξηζ , , ) = f ( σ)<br />
.<br />
Integrating with respect to σ ,<br />
<strong>and</strong><br />
ddd ξ η ζ = σ dσ,<br />
⎛<br />
⎞ f ( r) = f( σ) 4 sin( 2πνσ) sin( 2πνr)<br />
d<br />
ν<br />
rσ<br />
σ<br />
σ= 0 ⎜<br />
⎝ ν=<br />
0<br />
⎠⎟<br />
σ=∞ ν=∞<br />
⎜<br />
2 1<br />
2<br />
∫ ∫ d<br />
By changing <strong>the</strong> Summation order,<br />
2<br />
σ<br />
44
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
⎛<br />
⎞ ⎟<br />
f () r = 4 ⎜4 f()<br />
d ⎟<br />
⎝<br />
⎠<br />
( πνσ ) ( πνr<br />
)<br />
ν=∞ σ=∞<br />
sin 2 sin 2<br />
⎜<br />
2 2<br />
σ σ σ ∫ dν<br />
∫ ν . <br />
νσ<br />
νr<br />
ν= 0 ⎜ σ=<br />
0<br />
⎟<br />
Then, <strong>the</strong> 3-dimensional <strong>Fourier</strong>-<strong>Bessel</strong> Transform of f () r ,<br />
σ=∞<br />
sin(2 πνσ)<br />
4 ∫ f ( σ)<br />
σdσ,<br />
ν<br />
σ=<br />
0<br />
converges to a Hyper-real function<br />
F(2 πν)<br />
be infinite Hyper-reals, like <strong>the</strong> <strong>Delta</strong> <strong>Function</strong>.<br />
And <strong>the</strong> Inverse <strong>Fourier</strong>-<strong>Bessel</strong> Transform of F(2 πν)<br />
ν=∞<br />
sin(2 πνσ)<br />
4 ∫ F(2 πν)<br />
νdν<br />
σ<br />
ν=<br />
0<br />
converges to <strong>the</strong> Hyper-real function f ( ρ ).<br />
, some of its values may<br />
10.4 If f () r is Hyper-real function,<br />
Then,<br />
<strong>the</strong> Hyper-real integral<br />
σ=∞<br />
sin(2 πνσ)<br />
4 ∫ f ( σ)<br />
σdσ converges to<br />
ν<br />
σ=<br />
0<br />
F(2 πν)<br />
45
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
<strong>the</strong> Hyper-real integral<br />
ν=∞<br />
sin(2 πνσ)<br />
4 ∫ F(2 πν)<br />
νdν<br />
converges<br />
σ<br />
ν=<br />
0<br />
to f () r<br />
46
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
References<br />
[Bowman] Bowman, Frank, Introduction to <strong>Bessel</strong> <strong>Function</strong>s, Dover, 1958<br />
[Dan1] Dannon, H. Vic, “Well-Ordering of <strong>the</strong> Reals, Equality of all Infinities,<br />
<strong>and</strong> <strong>the</strong> Continuum Hypo<strong>the</strong>sis” in <strong>Gauge</strong> Institute Journal Vol.6 No 2, May<br />
2010;<br />
[Dan2] Dannon, H. Vic, “Infinitesimals” in <strong>Gauge</strong> Institute Journal Vol.6 No<br />
4, November 2010;<br />
[Dan3] Dannon, H. Vic, “Infinitesimal Calculus” in <strong>Gauge</strong> Institute Journal<br />
Vol.7 No 1, February 2011;<br />
[Dan4] Dannon, H. Vic, “Riemann’s Zeta <strong>Function</strong>: <strong>the</strong> Riemann Hypo<strong>the</strong>sis<br />
Origin, <strong>the</strong> Factorization Error, <strong>and</strong> <strong>the</strong> Count of <strong>the</strong> Primes”, in <strong>Gauge</strong><br />
Institute Journal of Math <strong>and</strong> Physics, November 2009.<br />
[Dan5] Dannon, H. Vic, “The <strong>Delta</strong> <strong>Function</strong>” in <strong>Gauge</strong> Institute Journal<br />
Vol.7 No 2, May 2011;<br />
[Dan6] Dannon, H. Vic, “The <strong>Delta</strong> <strong>Function</strong> <strong>and</strong> <strong>Fourier</strong> Transform” in<br />
<strong>Gauge</strong> Institute Journal Vol.7 No 3, August 2011;<br />
[Dirac] Dirac, P. A. M. The Principles of Quantum Mechanics, Second Edition,<br />
Oxford Univ press, 1935.<br />
[Gray] Gray, Andrew, <strong>and</strong> Ma<strong>the</strong>ws, G.B., A Treatise on <strong>Bessel</strong> <strong>Function</strong>s <strong>and</strong><br />
<strong>the</strong>ir applications to Physics, Dover 1966<br />
[Hen] Henle, James M., <strong>and</strong> Kleinberg Eugene M., Infinitesimal Calculus,<br />
MIT Press 1979.<br />
[Hosk] Hoskins, R. F., St<strong>and</strong>ard <strong>and</strong> Nonst<strong>and</strong>ard Analysis, Ellis Horwood,<br />
1990.<br />
47
<strong>Gauge</strong> Institute Journal<br />
H. Vic Dannon<br />
[Keis] Keisler, H. Jerome, Elementary calculus, An Infinitesimal Approach,<br />
Second Edition, Prindle, Weber, <strong>and</strong> Schmidt, 1986, pp. 905-912<br />
[Laug] Laugwitz, Detlef, “Curt Schmieden’s approach to infinitesimals-an eyeopener<br />
to <strong>the</strong> historiography of analysis” Technische Universitat Darmstadt,<br />
Preprint Nr. 2053, August 1999<br />
[Mikus] Mikusinski, J. <strong>and</strong> Sikorski, R., “The elementary <strong>the</strong>ory of<br />
distributions”, Rosprawy Matematyczne XII, Warszawa 1957.<br />
[R<strong>and</strong>] R<strong>and</strong>olph, John, “Basic Real <strong>and</strong> Abstract Analysis”, Academic Press,<br />
1968.<br />
[Riemann] Riemann, Bernhard, “On <strong>the</strong> Representation of a <strong>Function</strong> by a<br />
Trigonometric Series”.<br />
(1) In “Collected Papers, Bernhard Riemann”, translated from<br />
<strong>the</strong> 1892 edition by Roger Baker, Charles Christenson, <strong>and</strong><br />
Henry Orde, Paper XII, Part 5, Conditions for <strong>the</strong> existence of a<br />
definite integral, pages 231-232, Part 6, Special Cases, pages<br />
232-234. Kendrick press, 2004<br />
(2) In “God Created <strong>the</strong> Integers” Edited by Stephen Hawking,<br />
Part 5, <strong>and</strong> Part 6, pages 836-840, Running Press, 2005.<br />
[Schwartz] Schwartz, Laurent, Ma<strong>the</strong>matics for <strong>the</strong> Physical Sciences,<br />
Addison-Wesley, 1966.<br />
[Temp] Temple, George, 100 Years of Ma<strong>the</strong>matics, Springer-Verlag, 1981.<br />
pp. 19-24.<br />
[Watson] Watson, G. N., A Treatise on <strong>the</strong> Theory of <strong>Bessel</strong> <strong>Function</strong>s, 2 nd<br />
Edition, Cambridge, 1958.<br />
48