Complex Analysis Answers

𝑓 ^(𝑛) (𝑧) = 𝑛! /2𝜋𝑖 ∫_𝜕𝐷 𝑓(𝑤)𝑑𝑤 / (𝑤−𝑧)^𝑛+1 . using the generalized Cauchy integral formula (GCIF) expressed in

Prove it with the Mathematical Induction Method.(As is known, the formula for = 1 is Cauchy Integral

It becomes the Cauchy Integral Formula (CIF), which is the result of the theorem and it is correct.)

Show that

a)z+z*=2 Re z=2x

b)z-z*=2i Im z=2iy

c)z/z*={x^2-y^2/x^2+y^2}+i{2xy/x^2+y^2}

 𝐴 = { 𝑧 ∈ ℂ:|𝑧| < 1 𝑣𝑒 |𝑧 − 1/ 2 | > 1 /2 } ∪ { 1/ 2 } denote the set in the complex plane and

𝐴 ′ and 𝜕𝐴 = 𝐴̅ \ 𝐴𝑜

Write the set.

Let f(z) = sin z/z

and f(0) = 0. Explain why f is analytic at z = 0. Find the Maclaurian 

series for f(z) and g(z) = ∫ f(ξ)dξ from 0 to z

. Does there exist a function f with an 

isolated singularity at 0 and such that |f(z)|~ exp( 1/|z|) near z= 0?