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# hw1 - R 6 The Bessel function of zero order may be deFned...

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Homework 1 – Math 118C, Spring 2010 Due on Tuesday, April 6th, 2010 1. Prove that there exist constants γ > 0 and c > 0 such that vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle N summationdisplay n =1 1 n - log N - γ vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle c N . The number γ is called Euler’s constant , and γ 0 . 57721 · · · . 2. If n =0 a n x n has radius of convergence R , prove that it is uniformly convergent on every compact subset of the interval ( - R, R ). 3. Let f ( x ) = summationdisplay n =0 a n x n have radius of convergence R . Prove that then f exists, f ( x ) = summationdisplay n =1 na n x n 1 , and the radius of convergence is also R . 4. Let f ( x ) = summationdisplay n =0 a n ( x - x 0 ) n have radius of convergence R . Prove that f C (( x 0 - R, x 0 + R )), and a n = f ( n ( x 0 ) n ! . 5. Let

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Unformatted text preview: , R ]. 6. The Bessel function of zero order may be deFned by J ( x ) = ∞ s n =0 (-1) n x 2 n 4 n ( n !) 2 . ±ind its radius of convergence, and show that J is a solution of the di²erential equation xy ′′ + y ′ + xy = 0 . 1 2 7. Let y = f ( x ) be a solution of the diFerential equation x 2 dy dx-xy = sin x, with initial condition f (0) = c . ±ind a power series expansion for f near x = 0. 8. ±or what values of α and β does i ∞ dx x α + x β converge?...
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hw1 - R 6 The Bessel function of zero order may be deFned...

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