Help-me-complex-anaylsis-problem-f-z-is-entire-in-path-C-entire-Differantiable-complex-function-mean-f-z-satisfy-f-z-u-x-y-i-v-x-y-u-x-v-y-or-u-y-v-x-couch

Question Number 213862 by issac last updated on 19/Nov/24

Help me \dots . .!!! : (

complex anaylsis problem . .

f (z) is entire in path C

entire : Differantiable complex function

mean f (z) satisfy f (z) = u (x, y) + i \cdot v (x, y)

\frac{\partial u}{\partial x} = - \frac{\partial v}{\partial y} or \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} (couchy - riemann)

show that \int_{C} \frac{f (z)}{f^{'} (z)} d z = 2 π i \underset{h = 1}{\sum^{M}} P_{h} - Q_{h}

P_{h} is number of Zeros in path C

Q_{h} is number of poles in path C

and if pole is not exist

show that \frac{1}{2 π i} \int_{C} \frac{f (z)}{f^{'} (z)} d z

this equation is equivalent to the

formula for finding the number of zeros

f (z) = 0