Question

How to check whether function is analytic? f(z)=1/(z-z^5)

Answer 1

I know it's analytic if u_y= -v_x, and u_x=v_y. But is there a trick I can use where I don't have to expand z^5 where z=x+yi?

Answer 2

look for singularity, any region where it does not contain these singularites (poles) the function must be analytic.

Answer 3

dont understand what you mean by singularity

Answer 4

So if I factor my function out I get 1/z(1-z^4) how do I know it there are poles

Answer 5

that satisfy the equation

Answer 6

if the function is analytic in some region R
1# then it is infinitely differential
2# satisfies cauchy Riemann equations
3# closed loop integral is zero
4# the laurent expansion would consists of only +ve powers (analytic part) the principle part will be zero.

your have pole at 0,1,w,w^2,w^3,w^4 .. w is 5th root of unity.
for any region simply connected or multiply connected region that does not contain these points your function is analytic. just choose any given way you like to show it.

Answer 7

Thank you!