I am trying to compute the following integral:

$$ \int _0^\infty x\exp[-\pi b x^2]{_1F_1}[n,1,\frac{-\pi b x^2}{2}]{_2F_1}[1,\frac{2}{aq},1+\frac{2}{aq},\frac{-sx^{-a}}{2}]dx$$

Is there any general formula for computing integrals involving the confluent and the Gauss Hypergeometric functions with exponential functions?

Thank you in advance

PS: Note that this integral is equivalent to the following one:

$$ \int _{x=0}^\infty x\exp[-\pi b x^2]{_1F_1}[n,1,\frac{-\pi b x^2}{2}][\int_{t=0}^\infty e^{-t}t^{\frac{2}{aq}-1}{_1F_1}[1,1+\frac{2}{aq},\frac{-stx^{-a}}{2}]dt]dx$$ $$= \int _{t=0}^\infty e^{-t}t^{\frac{2}{aq}-1} [\int_{x=0}^\infty x\exp[-\pi b x^2]{_1F_1}[n,1,\frac{-\pi b x^2}{2}] {_1F_1}[1,1+\frac{2}{aq},\frac{-stx^{-a}}{2}]dx]dt$$

,i.e., by solving the inside integral, perhaps, we can reach to a simpler version of the original integral. Note also that the second confluent hypergeometric function can be further simplified to simple Gamma and incomplete Gamma functions.

  • $\begingroup$ Are you trying to find a definite integral? If so, what are the bounds of integration? $\endgroup$ Jul 22, 2013 at 17:02
  • $\begingroup$ I have updated the question. x=0->$\infty$ $\endgroup$
    – dioxen
    Jul 22, 2013 at 17:04

1 Answer 1


You can write the two hypergeometric functions as integrals (zero to one or zero to infinity). This integrals are integrals of powers of linear functions of the variables i.e. powers of polynomials. Then integrate to find another integral of polynomials. This will be another hypergeometric function. Another way is to expand in series both hypergeometric terms. Then change the variable of integration to square of x and rewrite the functions. You will see that the terms have an integral which can be evaluated by the gamma function. Then you normalize this gamma function to a Pochhammer symbol and try to recognize the new hypergeometric function. Because of the constants in the arguments of the two functions the new function is not single variable but multivariable hypergeometric function. Try Kampe de Feriet functions. If the case is not Kampe de Feriet it sure is GKZ alpha hypergeometric functions.


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