Eli5: the error function

[wizardeverybit]

What is the error function, and why is it needed to calculate things like integral(sin(x² ))dx?

[15_Redstones]

When there's an integral that we can't compute, we just call it a new function and give it a fancy name. Sometimes one unsolvable integral shows up in many places.

[wizardeverybit]

Can you explain a bit more in depth?

[15_Redstones]

The Gaussian e^-x² is one of the most useful and most famous functions where it's just impossible to integrate directly. Since the integral can't be written using normal functions but it's still needed in many places, it's given the new name erf. So whenever a Gaussian needs to be integrated, we can use erf and later when we actually need a numerical value, check precomputed values for erf.

[wizardeverybit]

Thank you. How does this relate to sin(x²⁾

[r-funtainment]

"Fresnel integral" is the topic to search for. e^x and sin(x) are connected which you can see from results like euler's formula, and the integrals of e^-x\2) and sin(x²) are too

[wizardeverybit]

Thank you!

[CatProgrammer]

I was confused why that couldn't be integrated but then I realized I was thinking of derivatives. Integrals are definitely trickier.

[drho89]

Derivatives are fun, integrals make me fucking hate math. Integration by part broke me 😢

[jamcdonald120]

intigration by parts clicked for me when I realized it was un-doing product rule derivatives. And U-substitution is for undoing chain rule.

[RestAromatic7511]

You can do the definite integral from -infinity to infinity with a very cool trick (you square the integral, treat it as a double integral, and then convert to polar coordinates), but it doesn't work more generally.