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Math Help - interchanging the order of integral and derivative

  1. #1
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    interchanging the order of integral and derivative

    could someone explain why the following equation is incorrect?

    \frac{d^2}{dx^2}\int^1 _{-1} \log|x-t|dt= \int^1_{-1}\frac{d^2}{dx^2}\log|x-t|dt= \int_{-1}^1 \frac{-1}{(x-t)^2}dt
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Kat-M View Post
    could someone explain why the following equation is incorrect?

    \frac{d^2}{dx^2}\int^1 _{-1} \log|x-t|dt= \int^1_{-1}\frac{d^2}{dx^2}\log|x-t|dt= \int_{-1}^1 \frac{-1}{(x-t)^2}dt
    Is \frac{\partial}{\partial t}\ln|x-t| even defined for t\in[-1,1] and x\in[-1,1]?
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    Is \frac{\partial}{\partial t}\ln|x-t| even defined for t\in[-1,1] and x\in[-1,1]?
    sorry i dont understand. the derivatives in the equation are with respect to x but not t.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Kat-M View Post
    sorry i dont understand. the derivatives in the equation are with respect to x but not t.
    Look here
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