# Cannot be obtained directly by the generalized power

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Unformatted text preview: ation (1.17). Let us focus now on this important case. If we set 1 a0 = p , 2 p! we obtain a2 = −a0 1 1 =− = (−1) 4p + 4 4(p + 1) 2p p! a4 = = 1 −a2 = 8p + 16 8(p + 2) 1 2(p + 2) 1 2 p+4 p+2 1 2 1 2 p+2 1 = 1!(p + 1)! 1 = (−1)2 1!(p + 1)! 1 2 and so forth. The general term is a2m = (−1)m 1 2 p+2m 24 1 , 1!(p + 1)! 1 . m!(p + m)! p+4 1 , 2!(p + 2)! 1 0.8 0.6 0.4 0.2 2 4 6 8 10 12 14 -0.2 -0.4 Figure 1.1: Graph of the Bessel function J0 (x). Thus we ﬁnally obtain the power series solution y= x 2 p∞ m=0 (−1)m 1 m!(p + m)! x 2 2m . The function deﬁned by the power series on the right-hand side is called the p-th Bessel function of the ﬁrst kind , and is denoted by the symbol Jp (x). For example, ∞ x 2m 1 J0 (x) = (−1)m . (m!)2 2 m=0 Using the comparison and ratio tests, we can show that the power series expansion for Jp (x) has inﬁnite radius of convergence. Thus when p is an integer, Bessel’s equation has a nonzero solution which is real analytic at x = 0. Bessel functions are so important that Mathematica includes them in its library of built-in fu...
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