Misplaced Pages

Faxén integral

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

In mathematics, the Faxén integral (also named Faxén function) is the following integral

Fi ⁡ ( α , β ; x ) = ∫ 0 ∞ exp ⁡ ( − t + x t α ) t β − 1 d t , ( 0 ≤ Re ⁡ ( α ) < 1 , Re ⁡ ( β ) > 0 ) . {\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).}

The integral is named after the Swedish physicist Olov Hilding Faxén, who published it in 1921 in his PhD thesis.

n-dimensional Faxén integral

More generally one defines the n {\displaystyle n} -dimensional Faxén integral as

I n ( x ) = λ n ∫ 0 ∞ ⋯ ∫ 0 ∞ t 1 β 1 − 1 ⋯ t n β n − 1 e − f ( t 1 , … , t n ; x ) d t 1 ⋯ d t n , {\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},}

with

f ( t 1 , … , t n ; x ) := ∑ j = 1 n t j μ j − x t 1 α 1 ⋯ t n α n {\displaystyle f(t_{1},\dots ,t_{n};x):=\sum \limits _{j=1}^{n}t_{j}^{\mu _{j}}-xt_{1}^{\alpha _{1}}\cdots t_{n}^{\alpha _{n}}\quad } and λ n := ∏ j = 1 n μ j {\displaystyle \quad \lambda _{n}:=\prod \limits _{j=1}^{n}\mu _{j}}

for x ∈ C {\displaystyle x\in \mathbb {C} } and

( 0 < α i < μ i , Re ⁡ ( β i ) > 0 , i = 1 , … , n ) . {\displaystyle (0<\alpha _{i}<\mu _{i},\;\operatorname {Re} (\beta _{i})>0,\;i=1,\dots ,n).}

The parameter λ n {\displaystyle \lambda _{n}} is only for convenience in calculations.

Properties

Let Γ {\displaystyle \Gamma } denote the Gamma function, then

  • Fi ⁡ ( α , β ; 0 ) = Γ ( β ) , {\displaystyle \operatorname {Fi} (\alpha ,\beta ;0)=\Gamma (\beta ),}
  • Fi ⁡ ( 0 , β ; x ) = e x Γ ( β ) . {\displaystyle \operatorname {Fi} (0,\beta ;x)=e^{x}\Gamma (\beta ).}

For α = β = 1 3 {\displaystyle \alpha =\beta ={\tfrac {1}{3}}} one has the following relationship to the Scorer function

Fi ⁡ ( 1 3 , 1 3 ; x ) = 3 2 / 3 π Hi ⁡ ( 3 − 1 / 3 x ) . {\displaystyle \operatorname {Fi} ({\tfrac {1}{3}},{\tfrac {1}{3}};x)=3^{2/3}\pi \operatorname {Hi} (3^{-1/3}x).}

Asymptotics

For x → ∞ {\displaystyle x\to \infty } we have the following asymptotics

  • Fi ⁡ ( α , β ; − x ) ∼ Γ ( β / α ) α y β / α , {\displaystyle \operatorname {Fi} (\alpha ,\beta ;-x)\sim {\frac {\Gamma (\beta /\alpha )}{\alpha y^{\beta /\alpha }}},}
  • Fi ⁡ ( α , β ; x ) ∼ ( 2 π 1 − α ) 1 / 2 ( α x ) ( 2 β − 1 ) / ( 2 − 2 α ) exp ⁡ ( ( 1 − α ) ( α α y ) 1 / ( 1 − α ) ) . {\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).}

References

  1. Olver, Frank W. J. (1997). Asymptotics and Special Functions. A K Peters/CRC Press. p. 332. doi:10.1201/9781439864548. ISBN 978-0-429-06461-6.
  2. Faxén, Hilding (1921). Einwirkung der Gefässwände auf den Widerstand gegen die Bewegung einer kleinen Kugel in einer zähen Flüssigkeit (PhD). Uppsala University.
  3. Paris, Richard Bruce (2010). "Asymptotic expansion of n-dimensional Faxén-type integrals". European Journal of Pure and Applied Mathematics. 3 (6). A K Peters/CRC Press: 1006–1031.
  4. Kaminski, David; Paris, Richard B. (1997). "Asymptotics via iterated Mellin–Barnes integrals: Application to the generalised Faxén integral". Methods and Applications of Analysis. 4 (3): 311–325. doi:10.4310/MAA.1997.v4.n3.a5.
Categories: