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Math Help - Complex analysis function.

  1. #1
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    Complex analysis function.

    What i have to do in these case?!

    Show that the function f(z)=lnz is differentiable(express z in polar form)

    and another question is

    Try to differentiate the trigonometric functon
    f(z)=cosz and f(z)=sinz using the exponential definition.

    Thanks,
    NaNa
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  2. #2
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    Quote Originally Posted by NaNa View Post
    What i have to do in these case?!

    Show that the function f(z)=lnz is differentiable(express z in polar form)
    In polar form, z= r e^{i\theta} so that f(z)= ln(r e^{i\theta}). The fact that any integer multiple of 2\pi can be added to \theta giving the same z makes this a multivalued function.

    and another question is

    Try to differentiate the trigonometric functon
    f(z)=cosz and f(z)=sinz using the exponential definition.

    Thanks,
    NaNa
    Well, the "exponential definitions" are cos(z)= \frac{e^z+ e^{-z}}{2} and sin(z)= \frac{e^z- e^{-z}}{2i}.
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  3. #3
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    Quote Originally Posted by NaNa View Post
    What i have to do in these case?!

    Show that the function f(z)=lnz is differentiable(express z in polar form)

    and another question is

    Try to differentiate the trigonometric functon
    f(z)=cosz and f(z)=sinz using the exponential definition.

    Thanks,
    NaNa
    The first, if it's differentiable, it's analytic and if it's analytic it satisfies the Cauchy-Riemann equations. So, using:

    Log(z)=\ln(r)+i\Theta

    use the Cauchy-Riemann equations so show it's analytic. For the second, just do what it said: express:

    \cos(z)=\frac{e^{iz}+e^{-iz}}{2} and just differentiate it with respect to z. Same dif for \sin(z)
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  4. #4
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    thanks a lot!!!
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  5. #5
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    How I use the Cauchy-Riemann here?!?!
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  6. #6
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    is ok I found it
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