Keyword Analysis & Research: airy function integral representation

stackexchange.com

https://math.stackexchange.com/questions/4462989/where-does-the-airy-function-integral-representation-come-from

Jun 01, 2022 · Consider a more general Fourier-like integral representation of a solution to Airy's equation Ψ ( x) = ∫ C V ( ω) e i ω x d ω where C is a complex contour whose properties are yet to be decided (but we think of going to infinity in both direction along certain rays). Using integration by parts we have Reviews: 1

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wolfram.com

https://functions.wolfram.com/Bessel-TypeFunctions/AiryAi/07/

Bessel-Type Functions AiryAi [ z] Integral representations (8 formulas) On the real axis (4 formulas) Contour integral representations (4 formulas)

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wikipedia.org

https://en.wikipedia.org/wiki/Airy_function

In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(x) and the related function Bi(x), are linearly independent solutions to the differential equation

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stackexchange.com

https://math.stackexchange.com/questions/2332536/integrate-airy-function-from-0-to-infty

Jun 22, 2017 · Integrate Airy function from 0 to ∞. From the integral representation of Airy function Ai(x) = ∫∞ − ∞dτ 2πexp( − iτx)exp( − iτ3 3), It is easy to see that ∫∞ − ∞dxAi(x) = 1. However, I am wondering how to find ∫∞ 0dxAi(x). From this website, the result of the above integral is 1 3.

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man.ac.uk

http://oldwww.ma.man.ac.uk/~gajjar/magicalbooks/methods/notest1_2se19.html

5.4 Integral represensations of Airy functions and suppose we seek a solution in the form y ( z) = ∫ c F ( s) e s z d s. Substitution into the equation shows that ∫ c ( s 2 - z) F ( s) e s z d s = 0. Integrate by parts then [ - F ( s) e s z] C + ∫ C ( s 2 F + d F d s) e s z d s = 0.

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springer.com

https://link.springer.com/article/10.1007/BF00944750

Integral representations are obtained for some products of the Airy functions Ai ( x ), Bi ( x ), and Ai ( xe ±πi/6 ). From these results it is then possible to evaluate certain definite integrals including a number of Mellin transforms. Download to read the full article text References

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spie.org

https://spie.org/samples/TT103.pdf

Equation (4.3) is referred to as the Airy integral and can be shown to be the solution to a homogeneous differential equation of the type d2y dx2. ¼ xy: (4.4) This equation is generally known as the Airy equation or the Airy differential equation. However, caution must be exercised in differentiating Eq.

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nist.gov

https://dlmf.nist.gov/9.5

Sep 10, 2019 · d t, ⓘ. Symbols: Ai ⁡ ( z): Airy function , π: the ratio of the circumference of a circle to its diameter , d x: differential , exp ⁡ z: exponential function , e: base of natural logarithm , i: imaginary unit , ∫: integral and z: complex variable. Source:

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