How to Find and Classify the Singular Point for f(z)?

[Swati Jain]

Homework Statement

Find and classify the singular point for
f(z) = 1/ ( sin z - sin a)
Where a is an arbitrary real constant.

Homework Equations

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

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 ??

 

[RUber]

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

 

[Swati Jain]

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

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.

 

[Samy_A]

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

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