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I'm trying to calculate an integral with respect to a complex value. I just want to know if I can estimate the integral using the residue theorem separately for the real and imaginary parts of the mentioned value or I cannot at all use this method here.

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The real or imaginary part on their own aren't analytic and so you can't apply results for which they are required to be analytic separately to the two.

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tnx for the answer....but what should i do in order to solve such a integral using residuen? –  Hosis Mar 19 '13 at 12:42
    
Identify the poles within the given curve and the corresponding residues (=coefficients of $(z-a_0)^{-1}$ in the Laurent series). –  Berci Mar 19 '13 at 12:45
    
can you please give me an example for a contour when the pole is in the form of a+b*i? –  Hosis Mar 19 '13 at 12:53
    
$$\frac{1}{2 \pi i} \oint_C \frac{\pi dz}{z - 0.5 i} = \pi$$ –  muzzlator Mar 19 '13 at 13:24
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