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

Reid, W. H.

doi:10.1007/s000330050052

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“…Products of two Airy functions and their integral transforms are investigated in many mathematical and physical papers (see, for example, [3][4][5][6][7][8]) while products of three or more Airy functions meet quite rarely. Besides the Laurenzi paper [1], these products are discussed in [9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Mellin transform of quartic products of shifted Airy functions

Abramochkin

Razueva

2016

Integral Transforms and Special Functions

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Section: Introductionmentioning

confidence: 99%

Mellin transform of quartic products of shifted Airy functions

Abramochkin

Razueva

2016

Integral Transforms and Special Functions

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Semi-integer derivatives of the Airy functions and related properties of the Korteweg-de Vries-type equations

Варламов¹

2007

Z. angew. Math. Phys.

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Fractional integrals and derivatives of Airy functions (Riesz potentials) are considered. For half integrals D −1/2 Ai(x) and D −1/2 Gi(x) explicit representations are found in terms of the products of Airy functions. Here Ai(x) and Gi(x) are the Airy function of the first kind and the Scorer function, respectively. Based on that representations are obtained for all semiinteger derivatives of Ai(x) and Gi(x). Applications to Korteweg-de Vries type equations are provided. (2000). 35C15, 33E20, 35Q53. Mathematics Subject Classification

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

Cited by 3 publications

References 4 publications

Mellin transform of quartic products of shifted Airy functions

Mellin transform of quartic products of shifted Airy functions

Semi-integer derivatives of the Airy functions and related properties of the Korteweg-de Vries-type equations

Fractional derivatives of products of Airy functions

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