# Calculating $\int_\gamma \frac{1}{1+z^2}\,dz$

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 $1+i$?

• Welcome to MSE! What've you tried? – Arpit Kansal Oct 2 '15 at 2:45
• I know that $\arctan z$ is a primitive for that function, but I'm not sure how to calculate it for complex values. – user276061 Oct 2 '15 at 2:45
• What do you mean by calculate arctan? Writ it using some more "elementary" function? Do you think that logarithms are simpler to work with? – mickep Oct 2 '15 at 3:59
• What's a primitive function? – Rudy the Reindeer Oct 2 '15 at 5:22
• @RudytheReindeer "primitive function" is sometimes used as a synonym for "antiderivative". – Tom-Tom Oct 2 '15 at 7:15

## 1 Answer

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.