MCS 351 ENGINEERING MATHEMATICS SOLUTION OF ...

v y = u x = cos x cosh(−y), −v x = u y = c − sin x sinh(−y) =⇒ v = − cos x sinh(−y) + h(x)
=⇒ v x = sin x sinh(−y) + dh
dx
=⇒ h(x) = c
=⇒ v = − cos x sinh(−y) + c.
When c = 1, u = sin x cosh(y)
v y = u x = cos x cosh(y), −v x = u y = c sin x sinh(y) =⇒ v = cos x sinh(y) + h(x)
=⇒ v x = − sin x sinh(y) + dh
dx
=⇒ h(x) = c
=⇒ v = cos x sinh(y) + c.
24. u = ax 3 + by 3 harmonic =⇒ u xx + u yy = 0.
u x = 3ax 2 , u y = 3by 2
=⇒ u xx = 6ax and u yy = 6by =⇒ u xx + u yy = 0 and ax + by = 0 =⇒ a = b = 0.
=⇒ u x = v y = 0 =⇒ v = h(x) =⇒ v x = dh
dx
=⇒ h(x) = c =⇒ v = c is constant.
13.5
1. e z = e x+iy = e x .e iy = e x (cos y + i sin y)
u = e x cos y and v = e x sin y =⇒ u x = e x cos y, u y = −e x sin y, v x = e x sin y and v y = e x cos y.
=⇒ u x = v y and u y = −v x .
=⇒ e z is analytic for every z. =⇒ e z is entire.
2-8 Values of e z :
2. z = 3 + πi ⇒ e 3+πi = e 3 (cos π − i. sin π) = −e 3 ∼ = −20, 086 and |e 3+πi | ∼ = 20, 086.
3. e 1+2i = e(cos 2 + i. sin 2) and |e 1+2i | = e √ cos 2 2 + i. sin 2 2.
4. e √ 2− π 2 i = e √2 [cos( −π
−π
2
) + i. sin(
2 )] = −e√2 i ∼ = −4, 11325i and |e √ 2− π 2 i | ∼ = 4, 11325.
5. e 7πi/2 = e 0 [cos(7πi/2) + i. sin(7πi/2)] = i and |e 7πi/2 | = |i| = 1.
6. e (1+i)π = e π (cos π + i sin π) = −e π ∼ = −23, 1407 and |e (1+i)π | ∼ = 23, 1407.
7. e 0,8−5i = e 0,8 .(cos(−5) + i. sin(−5)) = 2, 23.(0, 28 + i.0, 95) and |e 0,8−5i | ∼ = 2, 2.
8. e 9πi/2 = e 0 [cos(9πi/2) + i. sin(9πi/2)] = i and |e 9πi/2 | = |i| = 1.
9-12 Real and Imaginary parts: Let z = x + iy.
9. e −2z = e −2x−2yi = e −2x (cos(−2y) + i. sin(−2y))
⇒ Re(e −2z ) = e −2x (cos(2y) and Im(e −2z ) = −e −2x (sin(2y).
10. e z3 = e (x+iy)3 = e (x3 −3xy 2 )+(3x 2 y−y 3 )i = e (x3 −3xy 2) (cos(3x 2 y − y 3 ) + i. sin(3x 2 y − y 3 ))
⇒ Re(e z3 ) = e (x3 −3xy 2) cos(3x 2 y − y 3 ) and Im(e z3 ) = e (x3 −3xy 2) sin(3x 2 y − y 3 ).
11. e z2 = e (x+iy)2 = e x2 −y 2 +2xyi = e x2 −y 2 (cos(2xy) + i. sin(2xy)).
12. e 1 z = e 1
x+iy
= e x−iy
x 2 +y 2 = e x
x 2 +y 2 [cos(
y
) − i. sin( )].
x 2 +y 2 x 2 +y 2
17
y