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I am trying to take the derivative with respect to $a$ of some function $I(a)=\int_{0}^{\infty}f(a,x)dx$. I would like to make sure that I am using the Leiniz Integral Rule correctly. Various web sources indicate a set of conditions that must hold for $f(x,a)$ and $\frac{\partial f(x,a)}{\partial a}$ when integration is done over infinite region. From reading this source (see Theorem 10.3 on page 13) the conditions that $f(x,a)$ and $\frac{\partial f(x,a)}{\partial a}$ must obey are:

  1. $f(x,a)$ and $\frac{\partial f(x,a)}{\partial a}$ are continuous over $x\in[0,\infty)$ and around $a$ that we are interested in.

  2. There exists an integrable function (over $x$) $g(x)$ such that $|\frac{\partial f(x,a)}{\partial a}|\leq g(x)$.

  3. There exists an integrable function (over $x$) $h(x)$ such that $|f(x,a)|\leq h(x)$.

Integrable here means $\int_{-\infty}^{\infty}g(x)dx<\infty$.

However, another source seems to omit condition 3 above. I am wondering which source is correct. If there are "both correct", when is condition 3 necessary?

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1 Answer 1

up vote 2 down vote accepted

Those are sufficient but not necessary conditions. Basically they mimic Theorem 2 at the following link without any reference to measure theory or Lebesgue integration. Condition 3 from your source is basically saying that $f(x,a)$ is Lebesgue integrable.

http://planetmath.org/encyclopedia/DifferentiationUnderIntegralSign.html

I prefer the related approach (when possible) of writing a single integral as an iterated integral and then switching the order of integration. Justifying switching the order of integration is usually much easier.

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I think Theorem 2 in the link you posted works well for my problem, thank you! However, I just have one more clarification: by Theorem 2 in the link the conditions 2 and 3 in the statement of the theorem in my question can be loosened to hold only over the region $x\in[0,\infty)$. Does that seem right to you? –  M.B.M. Feb 26 '13 at 20:29

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