Complex Analysis Answers

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

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

Use Residue theorem to compute
the integral from 0 to 2 pi
[ Sin theta ] / [ 4+ sin (theta) + cos (theta) ] dtheta


Full Details Please

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

Decide whether the series is convergent or divergent

A)) ∑ [ 8^{ n+i.(2^-n) } ] / [ 9 ^ n ]

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B)) ∑ Conjugate all of this ( [ n+in+(n^i) ] / [ i ^ n ] )

B Decide whether these series is convergent or divergent .


1. ∑ (n+i sin n ) ^ {1/n}
2 . ∑ ( n^{in} ) , where n ^ in is taken in the 3π -branch

Hint : If you wish,You can use the fact
lim(n→∞) Zn=0 <--> lim(n→∞) |Zn| = 0

Decide if this sequence is convergent or divergent:
√ n√ n√ n√ [ Cos n + i sin n ]
, n≥ 2


Note :
Please more than method
the above means
,third times powered to ^1/n
third root of n

Show That
This power series
∑ n from 2 to infinity of
[ (-1)^{n} . z ^{2n+3} ] = [ (z^{7} ) / (1+z{^2} ) ] ; |z| <1

Is power series ∑ n from 2 to infinity of
[ (-1)^{n} . i ^ {2ni} . i ^(3i) ] Convergent ? If yes ,Compute It's Sum

Show that if
∑ Zn is convergent , then ∑ Zn ^ 2 is also convergent

please more than method

Decide whether this sequence is convergent or divergent:

Bn = [ Ln (2-n) ] / [ | n +i {square root (2) | ^ {2} ] ,
n≥ 3

Note : please Ln is principle value
Please more than method if could appreciated