Fresnel-Integral

As Fresnel integrals two improper integrals are in mathematics, particularly in the branch of analysis, called, named after the physicist Augustin Jean Fresnel.

Definition

The two integrals

Hot Fresnel integrals. They result from the Gaussian error integral using the Cauchy integral theorem.

History

Fresnel dealt around 1819 with these integrals. Euler considered in 1781 the more general integrals

And

Fresnel integrals in quantum mechanics

They also play an important role in quantum mechanics. The approach to derive the quantum mechanics path integrals, based on integrals of the form:

A practical formulation of the normalization constant is

Is a natural integer. For the integral

And is then called Fresnel integral. Integrals of this form appear in the Feynman path integrals derived from the Schrödinger equation.

From the Fresnel integral is a complex number whose real and imaginary parts are determined by results

Both integrals converge. The cosine integral is due to the symmetry of the cosine invariant under a change of sign of, the anti-symmetric sinusoidal changes sign. From the addition results with and and a case distinction for the signum function as a solution of the Fresnel integral

This also explains the normalization constant, which must be the inverse of the integral solution exactly, so that the overall expression is 1. In quantum mechanics, one chooses this out for pragmatic reasons and the idea that a wave function corresponds to a probability; So the integral of this function must be 1, since the particle is described finally somewhere.

Swell

• Reinhold Remmert, Georg Schumacher: Function Theory 1 5th edition, Springer -Verlag 2002, ISBN 3540590757, pp. 178f.
• Reinhold Remmert, Georg Schumacher: Function Theory 2nd 3rd edition, Springer - Verlag 2007, ISBN 3540404325, page 47