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Separate into real and imaginary parts of tanh-1 (x+iy).
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enter image description hereSolution:

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Given:tanh1(x+iy)

tanh1x=12log[(1+z)(1z)]

tanh1(x+iy)=12log[(1+x+iy)(1xiy)]

=12log[(1+x+iy)log(1xiy)]

=12[12log[(1+x)2+y2]+i.tan1(y1+x)]

=12[12log[(1x)2+y2]i.tan1(y1x)]

=12[12log[(1+x)2+y2](1x)2+y2]+i[tan1y1+x+tan1y1x]

=12[12log[(1+x)2+y2](1x)2+y2]+i.tan1[y1+x+y1x/1y1+x×y1x]

=12[12log((1+x)2+y2(1x)2+y2)+i.tan1(2y1x2y2)]

Separate real and imaginary parts,

Real parts: =14log((1+x)2+y2(1x)2+y2)

Imaginary parts: 12tan1(2y1x2y2)

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