---
tag: D0TQ9F2ZDW
kind: concept
name: Fresnel integrals
msc: ["33B20"]
sources:
  - url: https://en.wikipedia.org/wiki/Fresnel_integral
same_as:
  wikidata: Q580253
  wikipedia: Fresnel integral
---

## Summary

The Fresnel integrals $S(x) = \int_0^x \sin(t^2)\, dt$ and $C(x) = \int_0^x \cos(t^2)\, dt$ are integrals of [[sine and cosine|08WFZP6P89]] of a square.
They arise in the optics of diffraction near an edge, studied by Augustin-Jean Fresnel.
The plane curve $(C(t), S(t))$ is the Euler spiral, whose curvature grows linearly with arc length.
Both integrals tend to $\sqrt{\pi/8}$ as $x \to \infty$.

## Explanations

### L1 human

The function $\sin(t^2)$ oscillates faster and faster as $t$ grows.
Its area from $0$ to $x$ therefore wobbles less and less and settles down to a limit.
If one draws the point $(C(t), S(t))$ as $t$ increases, one gets a spiral that winds tighter and tighter around a point.
Road and railway designers use this spiral to pass smoothly from a straight line into a curve.