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Find and classify singularity?

  • #1

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 ??
 

Answers and Replies

  • #2
RUber
Homework Helper
1,687
344
What if you look at the solution set to the problem:
##\sin a = \sin z##.
 
  • #3
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.
 
  • #4
Samy_A
Science Advisor
Homework Helper
1,241
510
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})##
 

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