Find the domain of the Bessel function of order 0 defined by the following. Solution 8 Jo(x) = (-1)x2n (-1)x2n n = 0 2²n (n!)² Let an lim n→∞ = [22n (n!)²] Then we get the following. an+1 n (-1) + 1x2(n + 1) = lim 22(n+1)(n+1)!² n-→∞ = lim +27 +2 22n (n!)2 (-1)x2n 22n (n!)2 x2n = 0 n→ 22n + 2(n + 1)²(n!)² = lim n→∞ < 1 for all x Thus, by the Ratio Test, the given series converges for all values of x. In other words, the domain of the Bessel function Jo is (−∞, ∞) = R.

Find the domain of the Bessel function of order 0 defined by the following. Solution 8 Jo(x) = (-1)x2n (-1)x2n n = 0 2²n (n!)² Let an lim n→∞ = [22n (n!)²] Then we get the following. an+1 n (-1) + 1x2(n + 1) = lim 22(n+1)(n+1)!² n-→∞ = lim +27 +2 22n (n!)2 (-1)x2n 22n (n!)2 x2n = 0 n→ 22n + 2(n + 1)²(n!)² = lim n→∞ < 1 for all x Thus, by the Ratio Test, the given series converges for all values of x. In other words, the domain of the Bessel function Jo is (−∞, ∞) = R.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.5: Rational Functions

Problem 51E

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