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squaremeplz
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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 ?
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