查看“︁Böhmer積分”︁的源代码

{{No footnotes|time=2022-04-10T12:41:59+00:00}}
在數學中，'''Böhmer積分'''是一種特殊的積分

: <math>\displaystyle C(x,\alpha) = \int_x^\infty t^{\alpha-1}\cos(t) \, dt</math>
: <math>\displaystyle S(x,\alpha) = \int_x^\infty t^{\alpha-1}\sin(t) \, dt</math>

因此，菲涅耳积分可以用 Böhmer 积分表示为

: <math>\operatorname{S}(y) = \frac1{2}-\frac1{\sqrt{2\pi}}\cdot\operatorname{S}\left(\frac1{2},y^2\right)</math>
: <math>\operatorname{C}(y) = \frac1{2}-\frac1{\sqrt{2\pi}}\cdot\operatorname{C}\left(\frac1{2},y^2\right)</math>

[[三角积分|正弦积分]]和[[三角积分|余弦积分]]也可以用 Böhmer 积分表示

: <math>\operatorname{Si}(x) = \frac{\pi}{2}-\operatorname{S}(x,0)</math>
: <math>\operatorname{Ci}(x) = \frac{\pi}{2}-\operatorname{C}(x,0)</math>

== 参考 ==

* {{Cite book|last=Böhmer|first=Paul Eugen|title=Differenzengleichungen und bestimmte Integrale.|url=https://books.google.com/books?id=pD5tAAAAMAAJ|publisher=Leipzig, K. F. Koehler Verlag|language=de|year=1939|access-date=2022-04-10|archive-date=2022-04-10|archive-url=https://web.archive.org/web/20220410120504/https://books.google.com/books?id=pD5tAAAAMAAJ}}
* {{Cite book|last=Oldham|first=Keith B.|last2=Myland|first2=Jan|last3=Spanier|first3=Jerome|title=An Atlas of Functions|publisher=Springer Science & Business Media|date=2010|page=401|url=https://books.google.com/books?id=UrSnNeJW10YC&pg=PA401|access-date=2022-04-10|archive-date=2022-04-10|archive-url=https://web.archive.org/web/20220410121041/https://books.google.com/books?id=UrSnNeJW10YC&pg=PA401}}
[[Category:特殊函数]]