Complex Analysis Answers

Use Cauchy Integral Formula to evaluate

Integral on Curve [ e ^ (z+1) ] / [ (z-i) ( z^2 + (i-1)z-i)^3 ]

Note : 1))please the figure is close curve I cannot paint it by writing but these points (3i,i,-i,-2i,-1) inside the figure if you need it when find the singularity inside the curve
2))Also I need all singularity ( then sure only if inside curve i need all one with the working of cauchy integral , then plus it to find the total integral

Use Residue theorem to compute
Principle value
integral from ( - infinity to infinity )

[3x] / [ ( x^2 + 1 ) ^2 (x+2) ] dx

let ∑ Zn be A convergent series such that 0< Arg (Zn) < (Pi /2)

Show that
∑ ( Re ( Zn). Im (Zn) ) is also Convergent

1 )) Find the Radius of convergent of the power series
n from 0 to ∞ ∑ [2 ^ n / i ^ n] . { (z+ pi ) } ^n
2)) Determine whether ∞ is a singularity of f(z)=[z^2] + [2/(z^3)] - [ 2 ]
if its a singularity ,classify it

Note :
Please Radius is limit |an / an+1 | as n approach to infinity
and general for of power series is ∑ [ an . (z-center )^n]

let ∑ Zn be A convergent series such that 0< Arg (Zn) < (Pi /2)

Show that
∑ ( Re ( Zn). Im (Zn) ) is also Convergent

Find the value(s) of constant B such that :

integral on curve for [ (1)- (3z) + (2 B {z^4}) +( z^6) + (3 {z^7}) + ( 11 {z ^ 100} ] /[ z^5] dz =
integral on curve for [ e ^ {Bz} + 2 z ] / [ z^3 ] dz
where C is the unit circle oriented counterclockwise

1 )) Find the Radius of convergent of the power series
n from 0 to ∞ ∑ [2 ^ n / i ^ n] . { (z+ pi ) } ^n
2)) Determine whether ∞ is a singularity of f(z)=[z^2] + [2/(z^3)] - [ 2 ]
if its a singularity ,classify it

find the taylor series of f(z)= e^ { z^2 + 2z -3} about z=-1 along with its convergent neighborhood

Find the sum of the power series

n from 0 to ∞ ∑ {(-1)^n / (2n)! } . ( z- { pi/2} ) ^2
afterward compute the sum of the series
n from 0 to ∞ ∑ 1 / (2n)!

Let f(z) be an analytic function in the annulus 0 <|z| < R for some positive real number R,Whose laurent series (in this annulus) is given by

f(z) = n from -∞ to ∞ ∑ { (-1)^n / (n^2)! ] } . Z ^ { 5n - n^2 -1}
A)) What Kind of Singularity is z=0 for f(z) ?
B)) Compute integral on Curve for [ z ^ 24 . f(z) dz] ,where C is a counterclockwise simple path lying in the annulus enclosing z=0
C)) Calculate Res (f) in z=0
D)) Evaluate Integral on Curve for [ sin Z .f(z) dz] , where C : |z| = (R/2) oriented counterclockwise
Note : please not by limit