Complex Analysis

“ca: ch_02” — 2012/3/27 — 19:14 — page 42 — #2

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Complex Analysis

As we know that every complex number can be written in the polar form as z = re

iθ

,sow = f (z)

can also be expressed as

u + iv = f (re

iθ

)

⇒ f (z) = u(r, θ) + iv(r, θ)

For example, the function f (z) = z + z

2

can be written as

f (z) = (x + iy) + (x + iy)

2

⇒ f (z) = (x + x

2

− y

2

) + i(y + 2xy)

where u(x , y) = x + x

2

− y

2

and v(x, y) = y + 2xy

In polar form, the above function can be written as

f (z) = re

iθ

+ r

2

e

2iθ

⇒ f (z) = r

[

(cos θ + r cos2θ) + i(sin θ + r sin2θ)

]

where u(r, θ) = r cos θ + r

2

cos 2θ and v(r, θ) = r sin θ + r

2

sin 2θ.

2.2.1 Single-Valued and Multivalued Functions

A function f that