The Argument Principle in Complex Analysis

t

function

1

z


Δ arg f(z) ≈ -3.14159

Let

C

be a closed contour parameterized by

z=z(t)

in the range

a≤t≤b

, and

f(z)

a function meromorphic inside and on

C

. Define

Δarg(f)=arg(f(

t

2

))-arg(f(

t

1

))

for some

t

2

>

t

1

. The argument principle relates the change in argument of

f(z)

as

z

describes

C

once in the positive direction to the number of zeros and poles inside the contour. The change in argument for one complete circuit around

C

is given by

Δ

C

arg(f)=arg(f(b))-arg(f(a))

. The Argument Principle then states:

1

2π

Δ

c

argf(z)=Z-P

, where

Z

and

P

are the number of zeros and poles inside the contour, counting multiplicities.

This Demonstration shows

Δargf(z)

for six simple functions over a circular contour

C:z(t)=1.5

it

e

for

t

in the range

0≤t≤2π

. At

t=2π

,

Z-P

is calculated from the final value of

Δargf(z)

. The left pane shows the progress of

z(t)

over the contour. Zeros of the selected function are shown by red points and poles by blue points. In the case

z

3

(z-1)

, the pole has order of three and is therefore counted three times. The same would be done for zeros of higher order.