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How do I calculate $\int_\gamma \frac{1}{1+z^2}\,dz$, where $\gamma$ is the straight line from $1$ to $1+i$, using primitive functions?

I know that $\arctan z$ is a primitive for that function, but how to calculate it for complex values such as $1+i$?

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    $\begingroup$ Welcome to MSE! What've you tried? $\endgroup$ – Arpit Kansal Oct 2 '15 at 2:45
  • $\begingroup$ I know that $\arctan z$ is a primitive for that function, but I'm not sure how to calculate it for complex values. $\endgroup$ – user276061 Oct 2 '15 at 2:45
  • $\begingroup$ What do you mean by calculate arctan? Writ it using some more "elementary" function? Do you think that logarithms are simpler to work with? $\endgroup$ – mickep Oct 2 '15 at 3:59
  • $\begingroup$ What's a primitive function? $\endgroup$ – Rudy the Reindeer Oct 2 '15 at 5:22
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    $\begingroup$ @RudytheReindeer "primitive function" is sometimes used as a synonym for "antiderivative". $\endgroup$ – Tom-Tom Oct 2 '15 at 7:15
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You can parametrize $\gamma (t)=1+it$ as the line going from $1$ to $1+i$ then use $\int_{\gamma} f(z)=\int_a^b f(\gamma (t))\gamma'(t) dt$
Using the above parametrization you get $\int_0^1 \frac {idt}{1+(1+it)^2}=\arctan(1+i)-\pi/4$
You can get $\arctan(z)=\frac 12 i(\ln(1+iz)-\ln(1-iz))$
but that sounds harder than just putting $\arctan(1+i)$ into a computing software.

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