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Math Help - complex analysis: differentiability of a complex function?

  1. #1
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    complex analysis: differentiability of a complex function?

    Hi I'm having some trouble with showing a function is analytic at a point. If someone could help me with these questions I'd appreciate it.

    (a) Find all points where the function f(z)=(x^3+y^3+3y) + i(y^3-x^3+3y) is differentiable and compute the derivative at those points.
    my thoughts:
    using the cauchy riemann formulae i found that this funtion is differentiable at all points z=x+iy where x^2=y^2+1 or y^2=x^2-1 which, by taking the square root on both sides, is undefined at x=0 and hence the derivative of f at a point z=x+i(sqrt(x^2-1)) is f'(x+i(sqrt(x^2-1)))=3x^2-i3x^2. Is this correct? If not, then where am I going wrong.

    I'm having more difficulty with this next question:
    (b)Is the function in (a) analytic at any point? Justify your answer clearly.
    my thoughts:
    by the definition:A function f(z) is said to be analytic at a point z0 if there exists some ε-spherical neighborhood of z0 at all points of which f '(z) exists. I'm having a hard time using this definition to show that it is analytic at all points z=x+iy where x^2=y^2+1 or y^2=x^2-1 or am I wrong in thinking that it's analytic at these points? I just don't know how to approach this and reason it out.

    Thanks in advance for any help.
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  2. #2
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    Remember that the Cauchy Riemann equations are

    \displaystyle \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} AND \displaystyle \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.

    I think you'll find that you have evaluated the second of these equations incorrectly...
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  3. #3
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    I've done it over a couple of times and I still come up with the same answer
    {partial derivative u}/{partial derivative y} = 3y^2+3
    -{partial derivative v}/{partial derivative x} = -(-3x^2)= 3x^2

    hence 3y^2+3=3x^2 and so y^2+1=x^2

    where am I going wrong?
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  4. #4
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    It doesn't look like you're doing anything wrong. The function is differentiable on the hyperbola x^2-y^2=1.
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    Quote Originally Posted by chocaholic View Post
    I've done it over a couple of times and I still come up with the same answer
    {partial derivative u}/{partial derivative y} = 3y^2+3
    -{partial derivative v}/{partial derivative x} = -(-3x^2)= 3x^2

    hence 3y^2+3=3x^2 and so y^2+1=x^2

    where am I going wrong?
    Sorry, you are right, I misread what you wrote as you having found two separate equations that could satisfy the CR equations...
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    O ok sorry I should have stated that I was simplifying. So the differentiability part is correct but what about the point at which it is analytic?
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  7. #7
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    Quote Originally Posted by chocaholic View Post
    by the definition:A function f(z) is said to be analytic at a point z0 if there exists some ε-spherical neighborhood of z0 at all points of which f '(z) exists.

    As DrSteve said, there exists f'(z) iff z=x+iy satisfies H\equiv x^2-y^2=1 so, there is no \epsilon-spherical neighborhood of z_0\in H contained in H . Hence, f is nowhere analytic.


    Fernando Revilla
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