Moment integrals of powers of airy functions

Laurenzi, Bernard J.

doi:10.1007/bf00942815

Cited by 12 publications

(10 citation statements)

References 2 publications

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“…In order to prove (2.30) we recall the result of [16], 31) and the asymptotic expansion obtained in [23] for…”

Section: Proposition 4 the Integral Over R Of The Product Of W(x) Anmentioning

confidence: 99%

“…The products of Airy functions can also be considered special functions in their own right [2,21]. Integral representations for double, triple and quartic products of Airy functions were the subject of a series of papers [3,4,16,[21][22][23]27]. Integral representations for Ai 2 (x) and Ai(x)Bi(x), where Ai(x) and Bi(x) are the Airy functions of the first and second kind, respectively, first appeared in the papers [3,4] …”

Section: Introductionmentioning

confidence: 99%

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Fractional derivatives of products of Airy functions

Варламов¹

2008

Journal of Mathematical Analysis and Applications

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Fractional derivatives of the products of Airy functions are investigated, D α {Ai 2 (x)} and D α {Ai(x) × Bi(x)}, where Ai(x) and Bi(x) are the Airy functions of the first and second type, respectively. They turn out to be linear combinations of D α {Ai(x)} and D α {Gi(x)}, where Gi(x) is the Scorer function. It is also proved that the Wronskian W(x) of the system of half integrals {D −1/2 Ai(x), D −1/2 Gi(x)} and its Hilbert transform W(x) = −H W(x) can be considered special functions in their own right since they are expressed in terms of Ai 2 (x) and Ai(x)Bi(x), respectively. Various integral relations are established. Integral representations for D α {Ai(x − a)Ai(x + a)} and its Hilbert transform −H D α {Ai(x − a)Ai(x + a)} are derived.

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“…In order to prove (2.30) we recall the result of [16], 31) and the asymptotic expansion obtained in [23] for…”

Section: Proposition 4 the Integral Over R Of The Product Of W(x) Anmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fractional derivatives of products of Airy functions

Варламов¹

2008

Journal of Mathematical Analysis and Applications

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“…and this result should be compared with the one obtained by Laurenzi [4] by a different method, namely…”

Section: The Products Ai 3 (Z) and Ai 2 (Z)bi(z)mentioning

confidence: 99%

Integral representations for products of Airy functions ¶Part 2. Cubic products

Reid

1997

Z. angew. Math. Phys.

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Integral representations are obtained for some cubic products of the Airy functions Ai(z) and Bi(z). These integral representations are of the Laplace contour type but they involve the modified Bessel functions of order 1 6 . From these results it is then possible to evaluate a number of definite integrals involving such cubic products.Mathematics Subject Classification (1991). 33C10, 34A20, 34E05, 44A15.

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“…He used these results in his investigation of the Orr-Sommerfeld equation. The products of Airy functions are special functions in their own right and were the subject of numerous publications (see [2,3,5,6,8,20,21,32,33] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Differential and integral relations involving fractional derivatives of Airy functions and applications

Варламов¹

2008

Journal of Mathematical Analysis and Applications

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Various differential and integral relations are deduced that involve fractional derivatives of the Airy function Ai(x) and the Scorer function Gi(x). Several new Wronskian relations are obtained that lead to the calculation of a number of indefinite integrals containing fractional derivatives of the Airy functions. New fractional derivative conservation laws are derived for equations of the Korteweg-de Vries type.

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Moment integrals of powers of airy functions

Cited by 12 publications

References 2 publications

Fractional derivatives of products of Airy functions

Fractional derivatives of products of Airy functions

Integral representations for products of Airy functions ¶Part 2. Cubic products

Differential and integral relations involving fractional derivatives of Airy functions and applications

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