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Math Help - show that f(z) is an entire function

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    show that f(z) is an entire function

     f(z) = exp[(z)^2]
         = exp[(x+iy)^2]
         = exp[x^2-y^2]\cdot exp[(2xy)i]
          = exp[x^2-y^2]\cdot (cos(2xy)+isin(2xy))

    u= exp[x^2-y^2]\cdot cos(2xy)
    v= exp[x^2-y^2]\cdot sin(2xy)

    u_x= 2exp[x^2-y^2]\cdot (xcos(2xy)-ysin(2xy)) = v_y

    u_y= -2exp[x^2-y^2]\cdot (xsin(2xy)+ycos(2xy))= -v_x

    The partial derivatives of v and u exist, are continuous and satisfy the cauchy-reimann equations for the whole complex plane, therefore, f(z) is an entire function.

    is this answer sufficient? or is there a way to show that the equations hold for the entire complex plane.
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  2. #2
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    Quote Originally Posted by linalg123 View Post
     f(z) = exp[(z)^2]
         = exp[(x+iy)^2]
         = exp[x^2-y^2]\cdot exp[(2xy)i]
          = exp[x^2-y^2]\cdot (cos(2xy)+isin(2xy))

    u= exp[x^2-y^2]\cdot cos(2xy)
    v= exp[x^2-y^2]\cdot sin(2xy)

    u_x= 2exp[x^2-y^2]\cdot (xcos(2xy)-ysin(2xy)) = v_y

    u_y= -2exp[x^2-y^2]\cdot (xsin(2xy)+ycos(2xy))= -v_x

    The partial derivatives of v and u exist, are continuous and satisfy the cauchy-reimann equations for the whole complex plane, therefore, f(z) is an entire function.

    is this answer sufficient? or is there a way to show that the equations hold for the entire complex plane.
    Yes this is sufficient. Depending on your definitions another way is to show that the function has power series that has an infinite radius of convergence.

    f(z)=e^{z^2}=\sum_{n=0}^{\infty}\frac{z^{2n}}{n!}
    This power series converges for all z by the ratio test.
    So the function is analytic, or holomorphic, or entire.
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