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Math Help - Complex Analysis - Cauchy Riemann Equations + Laurent Expansions

  1. #1
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    Complex Analysis - Cauchy Riemann Equations + Laurent Expansions

    Hi all, i have a couple of questions on my complex analysis course that i cannot do and was wondering if you could help at all.



    QUESTION 1

    a) Suppose that the functions f(z) = u(x,y) + iv(x,y) and g(x) = v(x,y) + iu(x,y) are analytic in some domain D. Show that both u and v are constant functions


    b) Let f be a holomorphic function on the punctured disk D'(0,R) = {z in complex plane : 0 < |z| < R}, where R > 0 is fixed. Write down the formulae for c(subscript)n in the Laurent expansion

    f(z) = sum from n = minus infinity to infinity c(subscript)n . z^n (by this i mean cn multiplied by z^n)

    Using these formulae, prove that if f is bounded on D'(0,R), it has a removable singularity at 0.


    c) FInd the maximal radius R > 0 for which the function f(z) = (sin z)^-1 is holomorphic in D'(0,R), and find the principal part of its Laurent expansion about z(subscript)0 = 0 (centred at 0)
    (the z(subscript)0 refers to: sum from k = minus infinity to infinity c(subscript)k(z - z(subscript)0)^k the Laurent expansion)




    QUESTION 2

    Find the Laurent exansions of the function f(z) = z / (z^2 - 1) valid for:
    i) 0 < |z - 1| < 2
    ii) |z + 1| > 2
    iii) |z| > 1

    Thank you
    Doug
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  2. #2
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    Quote Originally Posted by dougwm View Post
    a) Suppose that the functions f(z) = u(x,y) + iv(x,y) and g(x) = v(x,y) + iu(x,y) are analytic in some domain D. Show that both u and v are constant functions
    Use the Cauchy-Riemann equations,
    u_x = v_y, u_y = -v_x, v_x = u_y, and v_y = -u_x.
    Thus,
    v_y = - v_y, u_x=-u_x, v_x = - v_x, and u_y = -u_y.
    Thus, \nabla u = \nabla v = \bold{0}.
    Thus, if D is a region (open connected set) then u,v are constant.
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    Quote Originally Posted by dougwm View Post
    b) Let f be a holomorphic function on the punctured disk D'(0,R) = {z in complex plane : 0 < |z| < R}, where R > 0 is fixed. Write down the formulae for c(subscript)n in the Laurent expansion

    f(z) = sum from n = minus infinity to infinity c(subscript)n . z^n (by this i mean cn multiplied by z^n)

    Using these formulae, prove that if f is bounded on D'(0,R), it has a removable singularity at 0.
    The theorem says that we can write,
    f(z) = \sum_{-\infty}^{\infty}a_n z^n where a_n = \frac{1}{2\pi i} \oint \limits_{|z|=R} \frac{f(\mu)}{\mu^{n+1}} d\mu.
    But we do not need this to prove that if f is bounded then it has a removable singularity.
    If |f|\leq A then |zf(z)| \leq A|z|.
    But \lim_{z\to 0}A|z| = 0 thus, \lim_{z\to 0}|zf(z)| = 0.


    c) FInd the maximal radius R > 0 for which the function f(z) = (sin z)^-1 is holomorphic in D'(0,R), and find the principal part of its Laurent expansion about z(subscript)0 = 0 (centred at 0)
    (the z(subscript)0 refers to: sum from k = minus infinity to infinity c(subscript)k(z - z(subscript)0)^k the Laurent expansion)
    If z\not = 0 then (\sin z)^{-1} is defined as long as z\not = \pi k.
    Thus, this radius is R=\pi.
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  4. #4
    Super Member wingless's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    If |f|\leq A then |zf(z)| \leq A|z|.
    But \lim_{z\to 0}A|z| = 0 thus, \lim_{z\to 0}|zf(z)| = 0.
    But the formula will be used to prove that

    "If \lim_{z\to z_0} (z-z_0)f(z) = 0 then z_0 is a removable singularity."


    In order to prove this we have to show that all a_{-n}'s are 0 (this is the definition of removable singularity).

    a_n = \frac{1}{2\pi i} \int \limits_{\gamma_r} f(w)(w-z_0)^{-n-1}~dw

    a_{-n} = \frac{1}{2\pi i} \int \limits_{\gamma_r} f(w)(w-z_0)^{n-1}~dw

    ( \gamma_r is a circle of radius r centered at z_0)

    We know that f is bounded, |f(w)(w-z_0)| \leq \epsilon,

    |f(w)(w-z_0)| \leq \epsilon \quad \rightarrow \quad  |f(w) \cdot r| \leq \epsilon \quad  \rightarrow \quad  |f(w)| \leq \frac{\epsilon}{r}

    Now we can use the formula.

    |a_{-n}| = \left |\frac{1}{2\pi i} \int \limits_{\gamma_r} f(w)(w-z_0)^{n-1}~dw \right | \leq (\frac{1}{2\pi})(2\pi r) (\frac{\epsilon}{r}r^{n-1}) = \epsilon r^{n-1}

    And \epsilon r^{n-1} goes to 0 for smaller circles.
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    Thank you both very much for your quick replies and great responses. They have helped alot.

    For the question 2, about the Laurent expansions I have:


    z/(z^2 - 1)

    by partial fractions this becomes:

    1/2(z - 1) + 1/2(z + 1)

    so for:

    0 < |z - 1| < 2


    1/2(z + 1) = 1/2(2 + z - 1) = 1/4(1 + (z - 1)/2)

    = 1/4 x (sum from n to infinity) (-1)^n . [|z - 1|/2]^n

    this proves |z - 1| < 2

    not too sure how to prove that |z - 1| > 0 though.



    for:

    |z + 1| > 2 i have:

    1/2(z - 1) = 1/2(-2 + z + 1) = 1/2 . 1/(z + 1) . 1/(1 - 2/(z + 1))

    = 1/2 . 1/(z + 1) . (sum from n to infinity) [ 2/|z + 1|]^n

    which proves |z + 1| > 2



    That is what i have so far, i am not sure how to do the other half of the partial differential equation in both cases. To prove it totally, also a little unsure on the third part.

    Thank you once again
    Doug
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  6. #6
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    Quote Originally Posted by dougwm View Post
    For the question 2, about the Laurent expansions I have:

    z/(z^2 - 1)

    by partial fractions this becomes:

    1/2(z - 1) + 1/2(z + 1)

    so for:

    0 < |z - 1| < 2


    1/2(z + 1) = 1/2(2 + z - 1) = 1/4(1 + (z - 1)/2)

    = 1/4 x (sum from n to infinity) (-1)^n . [|z - 1|/2]^n

    this proves |z - 1| < 2

    not too sure how to prove that |z - 1| > 0 though.
    That's correct. You have found the series expansion of 1/(2(z+1)) (in powers of z-1) which converges when |z - 1| < 2. But don't forget to include the other partial fraction 1/(2(z-1)). This is a negative power of z-1, so it is only defined when z≠1, or in other words when |z - 1| > 0.

    Quote Originally Posted by dougwm View Post
    for:

    |z + 1| > 2 i have:

    1/2(z - 1) = 1/2(-2 + z + 1) = 1/2 . 1/(z + 1) . 1/(1 - 2/(z + 1))

    = 1/2 . 1/(z + 1) . (sum from n to infinity) [ 2/|z + 1|]^n

    which proves |z + 1| > 2
    Again, the calculation is correct, but it's not appropriate to say that it "proves" |z + 1| > 2. What you should asy is that this series converges when |z + 1| > 2. (Here too, don't forget to include the other part of the partial fraction expression.)

    For the third part of the problem, in the region |z| > 1, you need to get series expansions of both parts of the partial fraction expression:

     \frac1{2(z - 1)} + \frac1{2(z + 1)} = \frac1{2z}\left(\frac1{1-\frac1z} + \frac1{1+\frac1z}\right) = \ldots.
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  7. #7
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    Quote Originally Posted by wingless View Post
    But the formula will be used to prove that

    "If \lim_{z\to z_0} (z-z_0)f(z) = 0 then z_0 is a removable singularity."
    I think your approach was better than mine because it is like the poster asked.

    However, the proof of the theorem (which is quoted) does not need to proven with approximating the integral. Remember a point z_0 is a removable singularity if there exists a function g in the punctured neighborhood of z_0 such that f(z) = g(z) for all 0<|z-z_0|<r and g is analytic at z_0. Now to prove this define the function h(z) = (z-z_0)f(z) for 0<|z-z_0|<r and h(z_0)=0. Then clearly h is analytic on the punctured neighrhood and continous at z_0. It follows that h is analytic on the entire disk. Finally, define g(z)=h(z)/(z-z_0) for z\not = 0 and g(z_0)=h'(z_0). Then g is analytic on the entire disk and g(z) = h(z) for all 0<|z-z_0|<r. Thus, f has removable singularity.
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