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Can anyone help me finding the real and the imaginary part of the function $f(x)=\cot(x+i)$, where $i=\sqrt{-1}$?

EDIT: $x\in \mathbb R$.

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Did you try WolframAlpha: wolframalpha.com/input/?i=cot%28x%2Bi%29 –  tards Nov 9 '11 at 3:19
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You don't say whether $x$ is supposed to be real. –  GEdgar Nov 9 '11 at 3:29
    
@tards: That was amazing! I didn't know about wolframalpha, thanks! –  Kris Nov 9 '11 at 7:53
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1 Answer

If $x\in \mathbb R$, then you should be able to use the definition of complex sine and cosine, and then multiply both the numerator and denominator by the complex conjugate.

$$\begin{align*}\cot(x+i) &= \frac{\cos(x+i)}{\sin(x+i)}\\ &=\frac{\cos x \cosh 1 -i\sin x \sinh 1}{\sin x \cosh 1 + i \cos x \sinh 1}\end{align*}$$ ... and so on. Related: http://en.wikipedia.org/wiki/Trigonometric_functions#Relationship_to_exponential_function_and_complex_numbers

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