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Homework Help: Find and classify singularity?

  1. Oct 27, 2015 #1
    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. jcsd
  3. Oct 28, 2015 #2


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    Homework Helper

    What if you look at the solution set to the problem:
    ##\sin a = \sin z##.
  4. Oct 28, 2015 #3
    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.
  5. Oct 28, 2015 #4


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    Science Advisor
    Homework Helper

    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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