Sketch the domain where it is analytic

[squaremeplz]

Homework Statement

determine if

a) f(z) = e^z / (z^2 + 4)

b) f(z) = conj(z) / |z|^2

c) f(z) = sum from 0 to inf. [ (e^z / 3^n) * (2z - 4)^n ]


is analytic and sketch the domain where it is analytic.

Homework Equations

The Attempt at a Solution

a) i don't know how to separate the function into a real and imaginary part. I have a feeling that the denominator needs to be manipulated but I have no clue how.

f(z) = e^(x + yi) / ((x+yi)^2 + 4)

f(z) = (e^x * cosy) / ((x+yi)^2 + 4) + (ie^x * siny) / ((x+yi)^2 + 4)

(e^x * cosy)* ((x-yi)^2 + 4) / ((x+yi)^2 + 4)((x-yi)^2 + 4)I'm guessing it is analytic everywhere except

z^2 = -4

b) f(z) = conj(z) / |z|^2

f(z) = 1 / z

1/z = 1 / ( x+yi)

(x - yi) / (x^2 + y ^2)

= x / (x^2 + y ^2) - yi / (x^2 + y ^2)

since du/dx u(x,y) = dv/dy u(x,y)

and dv/dx u(x,y) = - dv/dx u(x,y)

the function is analytic everywhere except at the origin.

c) I used the ratio test and got lim n-> inf |1/3 (2z - 4)| = |1/3 (2z - 4)|

it's analytic on 0 < |2z - 4| < 3 ?

 

[NJunJie]

some hint to start:
e^z is analytic.

by theroem: A rational function (the quotient of two polynomials) is analytic, except at zeroes of the denominator.

So take care of Z^2 + 4 - > find the values at which the term become zero.
Sketch it on the complex plane.

By the way, should be bounded by some region when u say if you would like to take integral... if not you would know if the whole function is analytic anot. In my opinion, z^2+4 will fail to be analytic when it is set to zero.

hence cauchy integral formula should be used.

hope it helps

 

[squaremeplz]

helps a bit, yet. I am having trouble transforming the original equation from part a) into f(z) = u(x,y) + i v(x,y) before the cauchy equations can be used.

 

[NJunJie]

(1) replace z = x + jy.
(2) expand and simplify.

:)

You'll get it. It may gets abit complicated. Be careful on the steps.