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A solution to the Laplace equation
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[QUOTE="Ray Vickson, post: 6036456, member: 330118"] Your solution does seem to satisfy the boundary condition for ##y=0##. Using a small non-zero value ##y = 0.001## we can plot the value of the solution $$f(x,y) = \frac{4 x y}{\pi} \int_0^{\infty} \frac{u \ln(1+u)}{[(x-u)^2+y^2)] [(x+u)^2+y^2]}$$ as a function of ##x## and compare it with the graph of ##\ln(1+x)##. I did this using Maple, and the results are given below. I just left the integral unevaluated and let Maple compute it numerically, but the same result was obtained if Maple did it symbolically. The two graphs ##f(x,0.001)## in blue and ##\log(1+x)## in red coincide, and just look like a single blue plot. I also (separately) plotted ##f(x,0.001)_{\text{numerical}}## and ##f(x,0.001)_{\text{analytical}}## together, and they likewise coincide. [ATTACH=full]228901[/ATTACH] [/QUOTE]
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A solution to the Laplace equation
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