1550251515-Classical_Complex_Analysis__Gonzalez_

Integration

R^2 - r^2 {2" d'lj;

= 271" lo J(() !( - z!^2

since

( z (( -zz R^2 - r^2

( - z + ( -z = !( - zl^2 = !( - z!2

Next, adding (7.28-5) to (7.28-2), we have

f(z) = 2~ 12" !(() [ ( ~ z + z ~ (] d'lj;

= f(O) + 2_ {

2

" J(() 2iRrsin(B -'lj;) d'lj;

271" lo · !( - zl^2

since
( z z z
--+---=l+------
(-z z-( (-z (-z

503

(7.28-6)

(7.28-7)

Rrei(IJ-'efl)_Rre-i(IJ-'efl) _

1

2iRrsin(B-'lj;)

= 1 + !( - z!2 - + !( - z!2

Letting

f(z) = u(r, B) + iv(r, B)

J(() = u(R,'lj;) +iv(R,'lj;)

we get, by equating real parts in (7.28-6),

R2 - r2 {2" d'lj;

u(r, B) = 271" Jo u(R, 'ljJ) !( - z!2

and by equating imaginary parts in (7.28-7),

.. 1 12 " 2iRr sin( e -'ljJ)

iv(r,B)=iv(O)+- u(R,'lj;) I( 12 d'lj;

271" 0 - z

Addition of (7.28-8) and (7.28-9) yields Schwarz's formula,

(7.28-8)

(7.28-9)

f(z) = iv(O) + 2~ 12" u(R,'lj;) R2 - r2 ~t~R:,:in(B - 'lj;) d'lj;

1 127' ( + z

=iv(O)+ -

2

u(R,'lj;)-,.. -d'lj;

71" 0 .,,-z

since

(( + z)((-z) = R^2 - r^2 + 2iRrsin(B - '1/J)

and