please answer it all and just a and b from each question 1. Classify the singula

Question

please answer it all and just a and b from each question

1. Classify the singularities of the following functions: z+1 cosh z (a) sinh? 2. Calculate the residues at the indicated poles cos z sin z 2z dz, where C is the circle , using a suitable path param- C 22 +4 eterisation 4. Evaluate dz, where C is the circle 21, using Cauchy's integral formula. 5. Use Cauchy's residue theorem to evaluate the following integrals along the circle 6. Use closed contour integration to evaluate -6r 10 dr +13 7. Use integration along the unit circle lzl to evaluate 0 3+2 cos ' Jo 4+sin ?

Explanation / Answer

1. a)

Clearly z=0 is a singularity of the function f(z)= (sinhz)/z.

Now limz->O (z-0)•f(z)= limz->0 (z•sinhz/z)= limz->0 sinhz =0.

So z=0 is a removable singularity of the function f(z)=(sinhz)/z.

b)

Clearly z=0 is a singularity of f(z)= (z+1)/z^2.

Also f(z)= 1/z + 1/z^2, so that the Laurent's expansion of f(z) in a neighbourhood of z=0 has finitely many terms involving negative powers of z. So z=0 is a pole for f(z)= (z+1)/z^2.

c)

Clearly z=0 is a singularity of the function f(z)= (coshz)/z^3.

Also using coshz = (e^z+e^(-z))/2 we see that the Laurent's expansion of f(z) in a neighborhood of z=0 has finitely many terms involving negative powers of z (in fact the highest negative powered term is 2/z^3).

Therefore z=0 is a pole of f(z)=(coshz)/z^3.

d)

Clearly z=0 is a singularity of f(z)=e^(1/z).

Also the Laurent's expansion of f(z) in a neighborhood of z=0 is

f(z)= 1+ 1/z + 1/z^2 + 1/z^3 + •••, so that it has infinitely many terms involving negative powers of z. So z=0 is an essential singularity of the function f(z) = e^(1/z).

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