I am looking at the following integral $$ \int |x^s| ds, $$ where the integral goes from $\sigma -iT$ to $\sigma + iT$ (along $Re(s) = \sigma$). I am expecting that the integral is real because the function is real. Let $s = \sigma + it.$ However, if I do a change of variable so that I take the integral with respect to $t$, I get the following:
Since $s = \sigma + i t$, I get $ds = i dt$.
So the integral above becomes
$$
\int_{-T}^T x^{\sigma} \ i \ dt,
$$
and this gives me a purely imaginary number. But I thought the integral was supposed to be real... Could someone please explain to me what I am doing wrong here (or where there is an error in my logic)? Thank you very much!!
Your result is correct, and your expectation that the result would be real is wrong. That the integral over a real-valued function is real is only true for integration in $\mathbb R$, where the measure is also real-valued. In this line integral, while the integrand is real, you can informally regard $\mathrm ds$ as an infinitesimal increment along the path, and for this particular path this increment is purely imaginary. Looking at the definition of the complex line integral, you can see that this informal intuition corresponds to purely imaginary finite increments along the path in the Riemann sum.