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Here is the definition of the Bessel function I am starting with a definition as an integral.

$$ J_n(x) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{i n t - x \sin t} \, dt $$

Essentially we have computed the $n$-th Fourier coefficient of a certain function:

$$ e^{-ix \sin t} = \sum e^{int} J_n(x) $$

How can I show the integral satisfies the Bessel differential equation? And which theorem helps to justify differentiating under the integral sign?

$$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $$

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cactus314
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  • also http://math.stackexchange.com/questions/359107/show-that-bessel-function-j-nx-satisfies-bessels-differential-equation – cactus314 Feb 07 '16 at 18:16
  • Differentiating under the integral sign "can be derived using the fundamental theorem of calculus". In this particular case, since the limits of integration are constant, the first two terms would disappear and you are left with a partial derivative inside the integral. https://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign – cnick Feb 07 '16 at 18:44

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