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Math Help - Complex Variable

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    Member Jason Bourne's Avatar
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    Complex Variable

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    \mid cosz \mid^2 = cosh^2 y - sin^2 x ,   ~~~~        z=x + iy
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    Quote Originally Posted by Jason Bourne View Post
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    \mid cosz \mid^2 = cosh^2 y - sin^2 x ,   ~~~~        z=x + iy
    \cos (x+iy) = \cos x \cos (iy) - \sin x \sin (iy) = \cos x \cosh y - i \sin x \sinh y

    That means, |\cos (x+iy)|^2 = \cos^2 x \cosh^2 y + \sin^2 x \sinh^2 y

    But, \cosh^2 y \cos^2 x + \sin^2 x\sinh^2 y = \cosh^2 y (1-\sin^2 x)+\sin^2 x \sinh^2 y = \cosh^2 y - \sin^2 x (\cosh^2 y -\sinh ^2 y) = \cosh^2 y - \sin^2 x
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