Solving problems of the form $x^c - c^x = d$ in the complex plane.

Question

Is there a known procedure for solving for $x$ in $x^c - c^x = d$ with known $c, d \in \mathbb C$?

Answer 1

As Henning Makholm commented, I do not think that you could find analytical solutions if $d\neq 0$ and only numerical methods would help.

In the case where $d=0$, the general solution is given in terms of Lambert function which is multi-valued; it writes $$x=-\frac{c }{\log (c)}W\left(-\frac{\log (c)}{c}\right)$$ For example, using $c=3$ gives four solutions; excluding the trivial $x_1=3$, the other solutions are given by $$x_2=-\frac{3 W\left(-\frac{\log (3)}{3}\right)}{\log (3)}\approx 2.47805$$ $$x_3=-\frac{3 W\left(\frac{1}{3} (-1)^{1/3} \log (3)\right)}{\log (3)}\approx -0.552925 - 0.60106 i$$ $$x_4=-\frac{3 W\left(-\frac{1}{3} (-1)^{2/3} \log (3)\right)}{\log (3)}\approx -0.552925 + 0.60106 i$$

You can easily concieve that, if $c$ is a complex, the solutions will still be more complicated.

For example, if $c=1+i$, the solution is given by $$x=-\frac{(1+i) W\left(\left(-\frac{1}{2}+\frac{i}{2}\right) \log (1+i)\right)}{\log (1+i)}$$ and you see the multi-value problem with the logarithm of complex numbers.