Hi All,(adsbygoogle = window.adsbygoogle || []).push({});

I met the following function to evaluate,

$$v(x)=\int_{-\infty}^{-a} G(t) ln(\vert t - x \vert) \mathrm{d}t + \int_{-a}^{a} -N ln(\vert t - x \vert) \mathrm{d}t + \int_{a}^{\infty} G(t) ln(\vert t - x \vert) \mathrm{d}t$$, where G is an unknown even function, N is a constant.

After multiple attempts I discovered I am not even close to the solution, which is achieved by the change of variables $$t = ar \qquad x = a \zeta \qquad g(r) = G(ar)/N $$, leading to the solution

$$ aN \{2ln (a) \int_{1}^{\infty} g(r) \mathrm{d}r + \int_{1}^{\infty} g(r) ln (\vert r^{2} - \zeta^{2} \vert ) -2 ln (a) -ln (\vert \zeta^{2}-1 \vert) + \zeta \, ln (\frac{\zeta - 1}{\zeta + 1}) +2 \}$$,

wellI get only the first term, but am completely lost as to where the squared variables in the logarithm argument came from in the second term, where could they come from...

Any hint would be the most welcome.

thanks as usual

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# Can not get Integral Right

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