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.

  • $\begingroup$ Please add copy/pastable Mathematica code for the expression and constraints. $\endgroup$ May 15 at 15:49
  • $\begingroup$ Dear Daniel, I did it. $\endgroup$ May 15 at 20:42
  • $\begingroup$ 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. $\endgroup$ May 15 at 21:29
  • $\begingroup$ 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. $\endgroup$ May 15 at 22:06
  • $\begingroup$ 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. $\endgroup$ 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]]

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


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


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


Mathematica fails (So do I.) with

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


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]]


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



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.