# Solving a complex integral using only the definiton

I'm struggling to solve such an integral using only the definition of the integral of complex function, any hints?

$$\int_{\gamma} \frac{dz}{z^2+4iz}.$$

contour $\gamma$ is a triangle with vertices -1+2i, -1-2i and 1

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Can you tell us that exact definition which shoud be used? – Berci Oct 15 '12 at 21:55
This means that I shouldn't use Newton-Leibniz formula or Cauchy formula, just plain integration – Jaqcues Oct 15 '12 at 22:04
It's hopeless without at least the Newton-Leibniz.. – Berci Oct 15 '12 at 22:11
Well, unless there's some rather nice trick I'm missing, trying to do this integral by means of parametrization and line integrals is going to be a huge pain in the mathematical gland. – DonAntonio Oct 15 '12 at 22:13
Could someone describe the solution process when using the Newton-Leibniz formula? How to decide which Ln branch to choose? – Jaqcues Oct 16 '12 at 7:00

By Cauchy's Residue theorem: since the function $\,\displaystyle{f(z)=\frac{1}{z^2+4iz}}\,$ only has the pole $\,z=0\,$ inside the domain enclosed by the triangle's perimeter and the function's analytic over the perimeter, we get
$$Res_{z=0}(f)=\lim_{z\to 0}zf(z)=\frac{1}{4i}\Longrightarrow\int_\gamma\frac{dz}{z^2+4iz}=2\pi i\frac{1}{4i}=\frac{\pi}{2}$$