1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

620 ANSWERS

7. W = 2(z + l)~ + 1-0gl-(z+ l)! + i7r.
   1+(%+1)!

(^9) • w -- l tr Arcsinz + i fr Arcsin .! ~ + !±i 2'
Section 1 2.l. Fourie r Series: page 520
00

1. U(t) = ~ I: 2 ;:_ 1 sin((2j - l )t].

j=l

00 00

3. V'(t) = ~ I: ( 21 ,: 1 >1 fi cos[(2j - l)t) = -;,^4 I: 2 ;:_ 1 sin[(2j - l)tJ = - U(t).

j=l J=l

00 00 00

5. ~ = V(O) = ~ ?:: ( 2 .,: 1 )2 cos(O] = * I: ( 2 ;,:i)', now solve for I: ( 2 ;,: 1 )2.

J=l J J=I J=l

7. U(t) = ~ f ~~/2; 1 ~~ sin[(2j - l )t).
   j=l
   00 00

9. U(t) = ~ I: 2 ;: 1 sin((2j - l )t] - * I: 2 ( 2 J 1 ) sin(2(2j - l)tJ. where an= 0

J=I J=I

for all n, and b4n = 0 for all n.

00 00

11. U(t) = ~ ?:: d-i sin[(2j - l)tJ + ~ ?:: 2 c 2 J_ 1 ) sin[2(2j - l)t]. where an= 0
    J=l 3=1

for all n, and b4n = 0 for all n.

Section 12.2. The Dirichlet Problem for the Unit Disk:

page 527

00

1. u(rcosB,rsinB) =~I: z;:_ 1 r^2 i-^1 sin((2j - l)B].

j=l

00 '-l'J- 1.

3. u (r cos B, r sin B) = ~ I: ~r^2 1-t cos((2j - 1)8].
   j=l
   00

5. u (r cos B, r sin B) =^3 ; + ~ I: (zj~t)2r^2 i-l cos[(2j - l)B]

J=l

00

- ~?:: 22 ( 2 J_ 1 p r^4 i-^2 cos[2(2j- l)B).

J = l

00

7. u(rcosB,rsinB) = i +~I: ( 2 i~l) 2 r^2 i-^1 cos[(2j-l)B]
   J=l
   00
   +~?:: 2 2< 2 J_ 1 ),r^4 i -^2 cos[2(2j - 1)8].
   J=I