- Bendix Ii Bessel Functions Of Half Integral Order When The Order V Is Half An Odd Integer That Is Besse 1 (109.12 KiB) Viewed 21 times
bendix II. Bessel Functions of Half-Integral Order When the order v is half an odd integer, that is, ±, ±, ±₁,..., Bessel functions of the first and second kinds can be expressed in terms of the elementary functions sin x, cos x, and powers of x. To see this let's consider the case when V = 1. From (7) we have T() = T(1 + ¹) = T() = T(1 + 3) = J1/2(x) = In view of the properties the gamma function, I(1 + a) = al(a) and the fact that I() = √ the values of I(1 + + n) for n = 0, n = 1, n = 2, and n = 3 are, respectively, In general, Hence, J1/2(x) = (-1)" non!(1 + + n) T() = T(1 + 3) = {T(): n=0 T() = -√√ T) = T(1 + 3) = T(): T() = 32 V (-1)" (2n + 1)! 22n+1n! Series Solutions of Linear Differential Equations = -5₂3³√ = 5·4·3·2·1√ 2³4.2 T(1 + / + n) = 7.5! 262! X 2 VTT We leave it as an exercise to show that X 2 2n + 1/2 J1/2(x) = (2n + 1)! √√ 22n+¹n! J-1/2(x) = 7.6.5! 26.6.2! 2n+1/2 2 π.Χ. π.Χ. 2 Π.Χ. -VT n=0 The infinite series in the last line is the Maclaurin series for sinx, and so we have shown that VT = sin .x. COS X. = 5! 252! 7! 273! -VT. (-1)" (2n + 1)! x2n+1 (26) (27)