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Evaluate $$\int_C Log(z) dz$$ where $Log(z)$ is the principle branch of the complex logarithm (Arg$(z)\in(-\pi,\pi)$) and $C$ is the contour given by the horizontal line connecting $z=i$ to $z=i+1$, and then the vertical line connecting $z=i+1$ to $z=1$.

The only way I can think of to do this is to write $Log(z)=lnr+i\theta$, where $z=re^{i\theta}$, but then I have difficulties parameterizing the contour line in polar coordinates.

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Break it into horizontal and vertical components

$$ \int_C = \int_{C_h} + \int_{C_v} $$

On $C_h$, take $z = i + x$. On $C_v$, take $z = 1+iy$. Thus

$$ \int_C \log(z) dz = \int_0^1 \log(i+ x) dx + \int_1^0 \log(1+iy) (i dy) $$

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Note that $\log(z)$ is holomorphic on a neighborhood of your path, and has an antiderivatve $z\log(z)-z$ on that same neighborhood. So all you have to do is evaluate between the endpoints: $$ \left.\int_{C} \log(z)\,dz = (z\log(z)-z)\right|_{z=i}^{1}=-1-\left[i\frac{i\pi}{2}-i\right]. $$

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