COMPLEX ANALYSIS

Question 1

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```lim<sub>z–>zo</sub> f(z) = ♾️ <=> ?

Answer 1

lim 1/f(x) = 0

Question 2

lim f(z) = w<sub>0</sub> {where z –> ♾️} <=> ?

Answer 2

lim f(1/z) = w<sub>0</sub> {where z –> 0}

Question 3

CAUCHY REIMANN EQUATION for Differentiablity

Answer 3

Suppose f(z) be a complex value function with f(z) = u+iv, if f is differentiatble at a point z<sub>o</sub> belongs to C, then

1. f<sub>y</sub> = if<sub>x</sub>
2. Ux = Vy and Uy = -Vx at z<sub>o</sub>
3. f’(z<sub>o</sub>) = Ux + iVx

Question 4

If e<sup>z</sup> = 1 then, z=?

Answer 4

z = 2nπi

Question 5

If e<sup>z</sup> = -1 then, z=?

Answer 5

z = (2n+1)πi

Question 6

If e<sup>z</sup> = i then, z=?

Answer 6

z = (4n+1)(π/2)i

Question 7

If e<sup>z</sup> = -i then, z=?

Answer 7

z = (4n+3)(π/2)i

Question 8

Continuity of a function

Answer 8

A function f(z) is continuous at point z° iff lim <sub>z–> z°</sub> f(z) exists and equals to f(z°).

|z-z°| < s. => |f(z) - f(z°)| < E