Find and classify singularity?

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1. Oct 27, 2015

Swati Jain

1. The problem statement, all variables and given/known data
Find and classify the singular point for
f(z) = 1/ ( sin z - sin a)
Where a is an arbitrary real constant.

2. Relevant equations
f(z) = 1/ ( sin z - sin a)
Where a is an arbitrary real constant.

3. The attempt at a solution
There will be infinite number of singularities of sin z = sin a
Put z' = z-a
Denominator can be written as sin z - sin a = sin ( z'+ a) - sin a
= sin a cos z' + cos a sin z' - sin a = sin a ( cos z' - 1) + cos a sin z'
= sin a ( -1/2! Z'^ 2 + 1/4! Z'^ 4 + ......) + cos a ( z' - z'^ 3/3! +......)
= cos a z' - 1/2! Sin a z'^2 -.........
Now how to define a and singularities ??

2. Oct 28, 2015

RUber

What if you look at the solution set to the problem:
$\sin a = \sin z$.

3. Oct 28, 2015

Swati Jain

I thought in this way too.. But I could not find how to find relation between z and a. It may be z = a + 2n pi, But I am not sure. And further I don"t understand that how to find singularity in this case. Thanks for your response.

4. Oct 28, 2015

Samy_A

Maybe you can use the formula for a difference of sines:

$\sin z - \sin a = 2\sin(\frac{z-a} {2})\cos(\frac{z+a} {2})$