Fresnel integral

As a Fresnel integrals are in mathematics , especially in the branch of analysis , two improper integrals designated after the physicist Augustin Jean Fresnel named.

definition

The two integrals

${\displaystyle \int _{-\infty }^{\infty }\cos(t^{2})\,\mathrm {d} t=\int _{-\infty }^{\infty }\sin(t^{2})\,\mathrm {d} t={\tfrac {1}{2}}{\sqrt {2\pi }}}$

are called Fresnel integrals. They result from the Gaussian error integral using Cauchy's integral theorem .

history

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

${\displaystyle \int _{-\infty }^{\infty }e^{(a^{2}-1)t^{2}}\cos(2at^{2})\,\mathrm {d} t={\frac {\sqrt {\pi }}{1+a^{2}}},\qquad -1\leq a\leq 1}$

and

${\displaystyle \int _{-\infty }^{\infty }e^{(a^{2}-1)t^{2}}\sin(2at^{2})\,\mathrm {d} t={\frac {a\,{\sqrt {\pi }}}{1+a^{2}}},\qquad -1\leq a\leq 1.}$

Fresnel integrals in quantum mechanics

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

${\displaystyle {\mathcal {F}}^{(j)}\equiv {\mathcal {N}}\int _{-\infty }^{\infty }\ \mathrm {e} ^{i\alpha \xi ^{2}}\xi ^{j}\mathrm {d} \xi \,.}$

A practical formulation of the normalization constant is ${\displaystyle {\mathcal {N}}}$

${\displaystyle {\mathcal {N}}\equiv {\sqrt {\frac {\alpha }{i\pi }}}}$,

${\displaystyle j}$is a whole natural number. For is the integral ${\displaystyle j=0}$

${\displaystyle {\mathcal {F}}\equiv {\mathcal {F}}^{(0)}\equiv {\mathcal {N}}\int _{-\infty }^{\infty }\ \mathrm {e} ^{i\alpha \xi ^{2}}\mathrm {d} \xi }$

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

The Fresnel integral results in a complex number, the real and imaginary parts of which are determined by

${\displaystyle \int _{-\infty }^{\infty }\cos(\alpha \xi ^{2})\,\mathrm {d} \xi ={\sqrt {\frac {\pi }{2\left|\alpha \right|}}}}$ and

${\displaystyle \int _{-\infty }^{\infty }\sin(\alpha \xi ^{2})\,\mathrm {d} \xi ={\sqrt {\frac {\pi }{2\left|\alpha \right|}}}\cdot \operatorname {sign} (\alpha )\,.}$

Both integrals converge. Due to the symmetry of the cosine, the cosine integral is invariant to a sign change from , the antisymmetric sine changes the sign . The addition results with and and a case distinction for the signum function as a solution of the Fresnel integral ${\displaystyle \alpha }$ ${\displaystyle {\sqrt {i}}=e^{i{\frac {\pi }{4}}}}$ ${\displaystyle -1=e^{i\pi }}$

${\displaystyle {\mathcal {F}}\equiv {\mathcal {F}}^{(0)}\equiv {\mathcal {N}}\int _{-\infty }^{\infty }\ \mathrm {e} ^{i\alpha \xi ^{2}}\mathrm {d} \xi ={\sqrt {\frac {\alpha }{i\pi }}}\cdot {\sqrt {\frac {i\pi }{\alpha }}}=1\,.}$

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

swell

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

Web links

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