(Solved): 5. Let \( f(t) \) be a once continuously differentiable function from the reals to the complex num ...

5. Let \( f(t) \) be a once continuously differentiable function from the reals to the complex number which is 1-periodic. (That is suppose that \( f(t+1)=f(t) \) for every \( t \).) Suppose that \( f(t) \neq 0 \) for any real \( t \). Show that \[ \int_{0}^{1} \frac{f^{\prime}(t)}{f(t)} d t \] is an integer multiple of \( 2 \pi i \).