# Error function (defined on the whole complex plane) is entire

julypraise

## Homework Statement

The wiki page says that error function $$\mbox{erf}(z) = \int_{0}^{z} e^{-t^{2}} dt$$ is entire. But I cannot find anywhere its proof. Could you give me some stcratch proof of this?

## The Attempt at a Solution

I've tried to use Fundamental Theorem of Calculus but as it is the line integral I couldn't use it.

clamtrox
You can just expand the integrand as a Taylor series, integrate by terms (since everything is finite) and then convince yourself that the resulting series converges everywhere.

julypraise
You can just expand the integrand as a Taylor series, integrate by terms (since everything is finite) and then convince yourself that the resulting series converges everywhere.

Ah.. frankly I'm not sure of how to expand by Taylor series... Is it by using Cauchy's Theorem? Could you give me any reference textbook where I can look up the theorem for this Taylor series expansion?

clamtrox
Ah.. frankly I'm not sure of how to expand by Taylor series... Is it by using Cauchy's Theorem? Could you give me any reference textbook where I can look up the theorem for this Taylor series expansion?

If you pick any analysis textbook at random, the odds of it containing a chapter on Taylor series is very very high. The extension to complex variables is straightforward.

You can also proceed by doing the standard proof that the function is holomorphic, by using the Cauchy-Riemann equations, but then you also have to consider whether the function has an analytic continuation to the entire complex plane. For example logarithm is holomorphic in its domain, but is not an entire function. That's why the Taylor series route is more straightforward: you can show that the series converges everywhere, which automatically shows you that the function is entire.

jackmell

## Homework Statement

The wiki page says that error function $$\mbox{erf}(z) = \int_{0}^{z} e^{-t^{2}} dt$$ is entire. But I cannot find anywhere its proof. Could you give me some stcratch proof of this?

## The Attempt at a Solution

I've tried to use Fundamental Theorem of Calculus but as it is the line integral I couldn't use it.

I don't see what's wrong with just differentiating it, showing the derivative is analytic throughout the complex plane, then concluding it's entire. That is, since

$$\frac{d}{dz} \text{erf}(z)=e^{-z^2}$$

and $e^{-z^2}$ is analytic, thus the error function is entire.