In an answer to a question I needed the following integral: $$ f(z):=\int\limits_0^\infty t\coth(zt)e^{-t^2}dt; $$ it represented deviation from modularity of some other function. However I noticed that this only works in one area and fails to work in another. After some confusion I discovered its source: it seems that $f(z)$ as defined suffers a discontinuity. For example, here is the result of a numeric calculation: $$ \begin{array}{r|rl} x&f(x+7i)&\approx\\ \hline .005&.482859&-.000025 i\\ .004&.482859&-.000020i\\ .003&.482859&-.000015 i\\ .002&.482859&-.000010i\\ .001&.482859&-.000005 i\\ 0&&(?)\\ -.001&-.482859&-.000005 i\\ -.002&-.482859&-.000010i\\ -.003&-.482859&-.000015 i\\ -.004&-.482859&-.000020i\\ -.005&-.482859&-.000025 i \end{array} $$ Discontinuity is evident in the real part; the imaginary part seems to be extendable to something continuous but non-differentiable along the imaginary axis. The integral seemingly should converge also for (purely) imaginary $z$ but I could not obtain any reliable numerical approximations with methods known to me.

It seems like $f(z)$ represents a many-valued analytic function, and the integral jumps from one branch of it to another.

Is there a name for this phenomenon?

Is it possible to modify the representation in such a way that it stays on the same branch?

How can I obtain an alternative representation of the same analytic function which would clarify many-valuedness and show the branching points?