# Limit of a complex function of a complex variable expressed in polar form

I've got the complex function

$$f(t;z,\mu) = \mathrm{e}^{-z\sinh t-\mu t}$$

where $$t=R \mathrm{e}^{i\theta},\theta \in [0,\pi/2]$$ The function depends on two parameters, $$z>0, \mu \in \mathbf{C}$$ with $$\mathrm{Re}(\mu) > 0, \mathrm{Im}(\mu) < 0$$

I want Mathematica to calculate the limit $$R \to \infty$$ of this function (it should be zero), but I don't know how to proceed. Any hint?

I tried with the straightforward instructions

In[1]:= int = Exp[-z*Sinh[t] - \[Mu] *t] /. t -> R*Exp[i\[Theta]]

Out[1]= E^(-E^i\[Theta] R \[Mu] - z Sinh[E^i\[Theta] R])

In[2]:= Limit[int, R -> \[Infinity],
Assumptions -> {{z > 0}, {Re[\[Mu]] > 0}, {Im[\[Mu]] <
0}, {\[Theta] > 0}, {\[Theta] < \[Pi]/2}}]

Out[2]= ConditionalExpression[\[Infinity], \[Mu] \[Element] Reals &&
E^i\[Theta] < 0 && z > 0]


obtaining a nonsensical conditional expression... How can the exponential of an imaginary number be less than zero - at least in the range I indicated in Assumpions? It looks like my assumptions are not taken into account.

• Please add copy/pastable Mathematica code for the expression and constraints. – Daniel Lichtblau May 15 at 15:49
• Dear Daniel, I did it. – Roberto Ricci May 15 at 20:42
• The real part of that sinh can oscillate (plot it with theta set to 4*E/7) so it would seem the limit does not exist in general. – Daniel Lichtblau May 15 at 21:29
• I don't see the point. The limit should exist for any fixed value of theta, so at most the final result could be a function of theta. But I do I get Mathematica help me explore this? This is exactly what I was asking. – Roberto Ricci May 15 at 22:06
• The point is that if the real part of the sinh oscillates and the oscillations increase without bound then there is no limit. Which is the gist of a response by @user64494. – Daniel Lichtblau May 16 at 15:05

The limit under consideration does not exist except the case \[Theta]=0. Indeed, specifying the parameters and correcting a syntax error (I*\[Theta] instead of i\[Theta]) , we have

\[Theta] = Pi/4; \[Mu] = 1 - I; z = 1;
int = Exp[-z*Sinh[t] - \[Mu]*t] /. t -> R*Exp[I*\[Theta]];
Limit[ComplexExpand[int], R -> \[Infinity]]


$$\underset{R\to \infty }{\text{lim}}\left(e^{\left(-\sqrt{2}\right) R-\cos \left(\frac{R}{\sqrt{2}}\right) \sinh \left(\frac{R}{\sqrt{2}}\right)} \cos \left(\sin \left(\frac{R}{\sqrt{2}}\right) \cosh \left(\frac{R}{\sqrt{2}}\right)\right)-i e^{\left(-\sqrt{2}\right) R-\cos \left(\frac{R}{\sqrt{2}}\right) \sinh \left(\frac{R}{\sqrt{2}}\right)} \sin \left(\sin \left(\frac{R}{\sqrt{2}}\right) \cosh \left(\frac{R}{\sqrt{2}}\right)\right)\right)$$

and

MaxLimit[ComplexExpand[Re[-z*Sinh[t] - \[Mu]*t /. t -> R*Exp[I*\[Theta]]]], R -> \[Infinity]]


$$\infty$$

MinLimit[ComplexExpand[Re[-z*Sinh[t] - \[Mu]*t /. t -> R*Exp[I*\[Theta]]]],R -> \[Infinity]]


$$-\infty$$

Mathematica fails (So do I.) with

MaxLimit[ComplexExpand[int], R -> \[Infinity]]


Appendix.

ClearAll[z, \[Mu]]; \[Theta] = Pi/4;
int = Exp[-z*Sinh[t] - \[Mu]*t] /. t -> R*Exp[I*\[Theta]];
MaxLimit[ComplexExpand[ Re[-z*Sinh[t] - (Re[\[Mu]] + I*Im[\[Mu]])*t /.
t -> R*Exp[I*\[Theta]]]], R -> \[Infinity]]


$$\infty$$

MinLimit[ComplexExpand[Re[-z*Sinh[t] - \[Mu]*t /. t -> R*Exp[I*\[Theta]]]],R -> \[Infinity]]


$$-\infty$$