2
votes
0answers
29 views

Properties and representations of the the rescaled complementary error function $\mathrm{erfcx}{z}$

Consider the rescaled complementary error function: $$ \mathrm{erfcx}(z) = {e^{z^2}} \left( {1-\mathrm{erf}(z)} \right) $$ $z \in \Bbb{C}$ which also has the following integral representation: $$ ...
7
votes
1answer
96 views

Calculating $\text{erf}^{-1}(z)$ for $z\in\mathbb{C}$

All the information I found about inverse error function $\text{erf}^{-1}(z)$ was about $z\in\mathbb{R}$. Also I found some Taylor expansions for it, but as the function is unbounded near $z=\pm1$, ...
0
votes
0answers
67 views

What is the limit of erf in $\infty+i\times\infty$?

What is the following limit: $\lim_{x \to \infty, y \to \infty} \mbox{erf}\left(x+iy\right)$ Wolfram alpha seems to give $1$, but here the unique answer seems to tell that $\mbox{erf}$ diverges. ...
1
vote
1answer
125 views

Limits of complex error and gamma functions in the complex plane?

What are the following one-sided limits in the complex plane (in the form $x+iy$): For the complex error function: $\lim_{x \to 0^+, y \to 0^+}\text{erf}\left(x+iy\right) = $ $\lim_{x \to +\infty, ...
1
vote
2answers
547 views

How to prove error function $\mbox{erf}$ is entire (i.e., analytic everywhere)?

How do I prove the error function $$ \mbox{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} dt. $$ is entire? Could you give me some scratch proof?