Is there a way to show original function of Erf function?

Questions
1

Hello,

For example, Erf[1/2] will outputs
Screenshot2024-01-19183751.png

How to make the result to be the original format, such as
Screenshot2024-01-19184007.png

And another question is that why Integrate[E^(-t^2), {t, 0, 1/2}] wouldn’t be a number? I think this formula can be integrated.

2

I tried FunctionExpand, but it does not do what you want:

Erf[1/2] // FunctionExpand

You can use a replacement rule instead:

erfRule = 
  Erf[x_] :> 2/Sqrt[\[Pi]] Inactive[Integrate][E^(-t^2), {t, 0, x}];
Erf[1/2] /. erfRule

The expression Integrate[E^(-t^2), {t, 0, 1/2}] is a number, of course. To check it you can use NumericQ, not NumberQ:

NumberQ[Integrate[E^(-t^2), {t, 0, 1/2}]]
NumericQ[Integrate[E^(-t^2), {t, 0, 1/2}]]
3

Thanks.

  1. What is the difference between NumberQ and NumericQ?
  2. Is there a way to simplify the Erf[1/2] without Integral symbol？

Regards,

4

From the documentation:

NumberQ[expr] returns False unless expr is manifestly a number (i.e. has head Complex, Integer, Rational, or Real)

It is meant as very fast test that does not waste time to go deep. Its name is unfortunate, because for example NumberQ[Pi] gives False.

NumericQ instead takes its time to investigate, and NumericQ[Pi] gives True.

I am not an expert, but I doubt that there is a simpler way to write Erf[1/2].

5

(2.) Is there a way to simplify the Erf[1/2] without Integral symbol？

Open in Wolfram Cloud ↗
6

Thanks.

From your post, there is a result for Erf[1/2], but is there a way to display it with fraction without Integral symbol?

Regards

7

When there is an integral, often there is a series.
Here is one way Mathematica derives what is found at

https://mathworld.wolfram.com/Erf.html

Open in Wolfram Cloud ↗
8

Thanks. It is a good way. Maybe there is no way to show the result with fraction.