1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

428 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS

2x
•EXAMPLE 11.4 Showthat,P(x,y)=Arctan 2 2
1

is harmonic in the

x +y -

disk lzl < 1.

Solution The results of Exercise 7(b), of Section 10.2, show that the function

. + 1 2 2 ·2
f (z) = ~ = -x - y 2 - i x 2
i - z x2 + (y -1) x2 + (y - 1)
is a conformal mapping of the disk lzl < 1 onto the right half-plane Re(w) > 0.
The results from Exercise 7(b), Section 5.2, show that the function

il>(u, v) = Arctan~ = Arg(u+ iv)

u

is harmonic in the right half-plane Re(w) > O.

parts of f(z), we write

Taking the real and imaginary

l -x2 - y2

u(x,y)= 2 ( 1 )2

x + y -

-2x

and v(x,y)= 2 ·

x^2 + (y - 1)