Faxén integral explained

In mathematics, the Faxén integral (also named Faxén function) is the following integral^[1]

\operatorname{Fi}(\alpha,\beta;x) = \int ^{inf ty} _{0} \exp( - t + xt ^{\alpha} )t ^{\beta-1} d t, ( 0 \leq \operatorname{Re}(\alpha) < 1 , \operatorname{Re}(\beta)> 0 ).

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

More generally one defines the

n -dimensional Faxén integral as^[3] I _{n(x)=λ} _{n\int} ^{inf ty} ^{0} … \int ^{inf ty} _{0} t ^{\beta _{1-1}} _{1} … t ^{\beta _{n-1}} _{n} e ^{- f(t _{1,...,t} _{n;x)}} d t _{1 …} d t _{n,} withf(t _{1,...,t} _{n;x):=\sum\limits} ^{n} ^{j =1} t ^{\mu _{j}} _{j} - xt ^{\alpha _{1}} _{1} … t ^{\alpha _{n}} _{n} and λ _{n:=\prod\limits} ^{n\mu} ^{j} for x \in \C and( 0 <\alpha _{i} <\mu _{i, \operatorname{Re}(\beta} _{i)>0,} i =1 , ... ,n). The parameter λ _{n} is only for convenience in calculations.

Properties

Let

\Gamma denote the Gamma function, then\operatorname{Fi}(\alpha,\beta; 0 ) = \Gamma(\beta), \operatorname{Fi}( 0 ,\beta;x) = e ^{x} \Gamma(\beta). For \alpha = \beta = \tfrac{ 1 }{ 3 } one has the following relationship to the Scorer function\operatorname{Fi}(\tfrac{ 1 }{ 3 },\tfrac{ 1 }{ 3 };x) =3 ^{2/3} \pi \operatorname{Hi}( 3 ^{-1/3} x).

Asymptotics

For

x\to inf ty we have the following asymptotics^[4] \operatorname{Fi}(\alpha,\beta; - x)\sim (\Gamma(\beta/\alpha))/(\alpha y ^{\beta/\alpha}) , \operatorname{Fi}(\alpha,\beta;x)\sim \left( (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).

References

1. Book: Frank W. J. . Olver . A K Peters/CRC Press . Asymptotics and Special Functions . 1997 . 332 . 10.1201/9781439864548.
2. Hilding. Faxén . PhD. . Einwirkung der Gefässwände auf den Widerstand gegen die Bewegung einer kleinen Kugel in einer zähen Flüssigkeit . 1921 .
3. Richard Bruce. Paris. A K Peters/CRC Press . Asymptotic expansion of n-dimensional Faxén-type integrals . European Journal of Pure and Applied Mathematics . 3 . 6 . 2010 . 1006–1031 .
4. David. Kaminski. Richard B.. Paris. Asymptotics via iterated Mellin–Barnes integrals: Application to the generalised Faxén integral . Methods and applications of analysis . 4 . 1997 . 311-325 .