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Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

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318 _Series_[C/i. 8<br />

Write the first three or four terms of a power-series expansion<br />

<strong>in</strong> x and y of the functions:<br />

2669. e x cosy.<br />

2670. (H-*) 1<br />

*'-<br />

Sec. 4. Fourier Series<br />

1. Dirichlet's theorem. We say that a function f (x) satisfies the Dirichlet<br />

conditions <strong>in</strong> an <strong>in</strong>terval (a, b) if, <strong>in</strong> this <strong>in</strong>terval, the function<br />

1) is uniformly bounded; that is \f(x)\^M when a < x < b, where M<br />

is constant;<br />

2) has no more than a f<strong>in</strong>ite number of po<strong>in</strong>ts of discont<strong>in</strong>uity and<br />

all of them are of the first k<strong>in</strong>d [i.e., at each discont<strong>in</strong>uity g<br />

the function f (x) has a f<strong>in</strong>ite limit on the left f (g 0)= Urn f (I e) and a<br />

~"<br />

f<strong>in</strong>ite limit on the right /(-{-0)= lim /( + e) (e>0)J;<br />

e ->o<br />

3) has no more than a f<strong>in</strong>ite number of po<strong>in</strong>ts of strict extrenium.<br />

Dirichlet's theorem asserts that a function /(*), which <strong>in</strong> the <strong>in</strong>terval<br />

( ji, Ji) satisfies the Dirichlet conditions at any po<strong>in</strong>t x of this <strong>in</strong>terval at<br />

which /(x) is cont<strong>in</strong>uous, may be expanded <strong>in</strong> a trigonometric Fourier series:<br />

f(x)=?+ a, cos x + b v s<strong>in</strong> x + a 2 cos 2x-\-b2 s<strong>in</strong> 2*+<br />

. . . +an cos nx +<br />

+ bn s<strong>in</strong>nx+..., (1)<br />

where the Fourier coefficients a n and b n are calculated from the formulas<br />

ji ji<br />

= \ f(x)cosnxdx(n = Q, 1, 2, ...);&= f<br />

JT J JI J<br />

-n -jt<br />

If x is a po<strong>in</strong>t' of discont<strong>in</strong>uity, belong<strong>in</strong>g to the <strong>in</strong>terval ( jt, n), of a<br />

function f (,v), then the sum of the Fourier series S (x) is equal to the arithme-<br />

tical mean of the left and right limits of the function:<br />

SM =~<br />

At the end-po<strong>in</strong>ts of the <strong>in</strong>terval * = n and X = K,<br />

2. Incomplete Fourier series. If a function / (*) is even [i. e., /(- x) =<br />

s=/(jc)], then <strong>in</strong> formula (1)<br />

and<br />

ji<br />

6 rt -0 (w = l f 2, ...)<br />

2 r<br />

a /"=^ J / = 0, 1,2, ...).

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