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TAMU MATH 311 - Airy-Bessel

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AIRY FUNCTIONS.Solutions of the equationy′′(x) − xy(x) = 0(note sign).Ai(x) =133√3Γ(2/3)−126√3Γ(2/3)πx +1183√3Γ(2/3)x3−1246√3Γ(2/3)πx4+15403√3Γ(2/3)x6−110086√3Γ(2/3)πx7+1388803√3Γ(2/3)x9+ Ox10and1Bi(x) =1335/6Γ(2/3)+1232/3Γ(2/3)πx +11835/6Γ(2/3)x3+12432/3Γ(2/3)πx4+154035/6Γ(2/3)x6+1100832/3Γ(2/3)πx7+13888035/6Γ(2/3)x9+ Ox102Our Airy functions: solutions of y′′+ xy = 0, with initial conditions y(0) = 1,y′(0) = 0 and y(0) = 0, y′(0) = 1.12Γ(2/3)31/6√3 Ai(−x) + Bi(−x)= 1 −16x3+1180x6−112960x9+11710720x12−1359251200x15+1109930867200x18−146170964224000x21+ O(x24).−13π31/3Γ(2/3)Bi(−x) −√3 Ai(−x)= x −112x4+1504x7+ O(x10).3Airy functions.12.5−1010.07.55.0x2.50.0−2.51050−5−1515.04Approximations by1 −16x3+1180x6−112960x9+11710720x12−1359251200x15+ O(x18)of order 3, 18, and 27.y4624−2−42x573150−1−33−510−2 −1−3326−254−12047351−40−5−31−2 −1−3xyy4624−2−42x573150−1−33−510−1−2−35Bessel functions J0and J1.y0.8160.48−0.4−0.80x1.0200.60.2120.0−0.2−0.64−1.0−4−8−12−16−20y0.8160.48−0.4−0.80x1.0200.60.2120.0−0.2−0.64−1.0−4−8−12−16−206Bessel functions J2and J3.y0.80.48−0.4−0.8160x1.0200.60.2120.0−0.2−0.64−1.0−4−8−12−16−2016y0.80.48−0.4−0.80x1.0200.60.2120.0−0.2−0.64−1.0−4−8−12−16−207First four Bessel


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TAMU MATH 311 - Airy-Bessel

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