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In mathematics, the Faxén integral (also named Faxén function) is the following integral[1]
![{\displaystyle \operatorname {Fi} (\alpha ,\beta ;x)=\int _{0}^{\infty }\exp(-t+xt^{\alpha })t^{\beta -1}\mathrm {d} t,\qquad (0\leq \operatorname {Re} (\alpha )<1,\;\operatorname {Re} (\beta )>0).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d165d7ed30fcc9f372a232a27379eaf52a39c1cf)
The integral is named after the Swedish physicist Olov Hilding Faxén, who published it in 1921 in his PhD thesis.[2]
n-dimensional Faxén integral[edit]
More generally one defines the
-dimensional Faxén integral as[3]
![{\displaystyle I_{n}(x)=\lambda _{n}\int _{0}^{\infty }\cdots \int _{0}^{\infty }t_{1}^{\beta _{1}-1}\cdots t_{n}^{\beta _{n}-1}e^{-f(t_{1},\dots ,t_{n};x)}\mathrm {d} t_{1}\cdots \mathrm {d} t_{n},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45053278064166136cd92d401d7d37d14cda2e64)
with
and ![{\displaystyle \quad \lambda _{n}:=\prod \limits _{j=1}^{n}\mu _{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3134952132ab1692fb6e817ab990420f75db700f)
for
and
![{\displaystyle (0<\alpha _{i}<\mu _{i},\;\operatorname {Re} (\beta _{i})>0,\;i=1,\dots ,n).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79bbb3dd7f25dcac7837ef2abe22301ab79ade15)
The parameter
is only for convenience in calculations.
Properties[edit]
Let
denote the Gamma function, then
![{\displaystyle \operatorname {Fi} (\alpha ,\beta ;0)=\Gamma (\beta ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a1840015e053c13c04324af826da0e1e4ea0f09)
![{\displaystyle \operatorname {Fi} (0,\beta ;x)=e^{x}\Gamma (\beta ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bf981d2843f70cd6b195813d9f558f2d7135da5)
For
one has the following relationship to the Scorer function
![{\displaystyle \operatorname {Fi} ({\tfrac {1}{3}},{\tfrac {1}{3}};x)=3^{2/3}\pi \operatorname {Hi} (3^{-1/3}x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97e6b0870342b43f5ee2372ad510ba3ac1dee7cb)
Asymptotics[edit]
For
we have the following asymptotics[4]
![{\displaystyle \operatorname {Fi} (\alpha ,\beta ;-x)\sim {\frac {\Gamma (\beta /\alpha )}{\alpha y^{\beta /\alpha }}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb67aa1f2a29df64ba8374d2c5403bb5f08f6c2f)
![{\displaystyle \operatorname {Fi} (\alpha ,\beta ;x)\sim \left({\frac {2\pi }{1-\alpha }}\right)^{1/2}(\alpha x)^{(2\beta -1)/(2-2\alpha )}\exp \left((1-\alpha )(\alpha ^{\alpha }y)^{1/(1-\alpha )}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a22fa3c680203a08ed624a9f1543627ebe2263a7)
References[edit]