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Thread: Complex integral help

  1. #1
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    Complex integral help

    C is the circle $\displaystyle |z|=a$, 'a' is a positive real number.

    $\displaystyle \displaystyle I=\frac{1}{2\pi i}\int_{C} \frac{z^2}{z^2+b} \ dz \ \ \ \ , \$ 'b' is a positive real number.

    I'm not sure how to evaluate integrals of the form I (given a and b). It looks like I need to use Cauchy's integral formula but I'm not sure if it is possible to use it here.

    Can someone help?
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  2. #2
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    Quote Originally Posted by 0-) View Post
    C is the circle $\displaystyle |z|=a$, 'a' is a positive real number.

    $\displaystyle \displaystyle I=\frac{1}{2\pi i}\int_{C} \frac{z^2}{z^2+b} \ dz \ \ \ \ , \$ 'b' is a positive real number.

    I'm not sure how to evaluate integrals of the form I (given a and b). It looks like I need to use Cauchy's integral formula but I'm not sure if it is possible to use it here.

    Can someone help?
    |z|= a is a circle about the origin with radius a. $\displaystyle z= ae^{i\theta}$ for any point on that circle, where $\displaystyle \theta$ is the angle the radius to the point z makes with the positive real (x) axis. Use that substitution and integrate as $\displaystyle \theta$ goes from 0 to $\displaystyle 2\pi$.

    You will have to consider the cases where $\displaystyle \sqrt{b}< a$ and $\displaystyle \sqrt{b}\ge a$ separately.
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  3. #3
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    Quote Originally Posted by 0-) View Post
    C is the circle $\displaystyle |z|=a$, 'a' is a positive real number.

    $\displaystyle \displaystyle I=\frac{1}{2\pi i}\int_{C} \frac{z^2}{z^2+b} \ dz \ \ \ \ , \$ 'b' is a positive real number.

    I'm not sure how to evaluate integrals of the form I (given a and b). It looks like I need to use Cauchy's integral formula but I'm not sure if it is possible to use it here.

    Can someone help?
    Cauchy's integral formula deals with integrals like $\displaystyle \frac{1}{2i\pi}\int_C \frac{f(z)}{z-z_0}dz$, right? This integral equals $\displaystyle f(z_0)$ if $\displaystyle z_0$ is inside the circle $\displaystyle C$, and 0 if $\displaystyle z_0$ is outside.

    You can write $\displaystyle \frac{z^2}{z^2+b^2}=1-\frac{b^2}{z^2+b^2}= 1-\frac{b^2}{(z+i\sqrt{b})(z-i\sqrt{b})}=$ $\displaystyle 1-\frac{b^2}{2i\sqrt{b}}\left(\frac{1}{z-i\sqrt{b}}-\frac{1}{z+i\sqrt{b}}\right)$ and use Cauchy's formula for each term.
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  4. #4
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    Thank you Laurent!

    I think your working is slightly wrong, all the $\displaystyle \sqrt{b}$'s should be $\displaystyle b$.

    So $\displaystyle \displaystyle \frac{z^2}{z^2+b^2} = 1-\frac{b}{2i}\left(\frac{1}{z-ib}-\frac{1}{z+ib}\right)$

    From this, I think I am right in saying that if I is the same integral in my previous post but with $\displaystyle b^2$ instead of $\displaystyle b$ then

    $\displaystyle |b| \leq |a|\implies I=0$

    Can someone confirm this for me?
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  5. #5
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    Quote Originally Posted by 0-) View Post
    Thank you Laurent!

    I think your working is slightly wrong, all the $\displaystyle \sqrt{b}$'s should be $\displaystyle b$.

    So $\displaystyle \displaystyle \frac{z^2}{z^2+b^2} = 1-\frac{b}{2i}\left(\frac{1}{z-ib}-\frac{1}{z+ib}\right)$
    Yes, there're mistakes in my post, but (for me) it was on the left-hand side: it should have been

    $\displaystyle \frac{z^2}{z^2+b}=1-\frac{b}{z^2+b}= 1-\frac{b}{(z+i\sqrt{b})(z-i\sqrt{b})}=$ $\displaystyle 1-\frac{b}{2i\sqrt{b}}\left(\frac{1}{z-i\sqrt{b}}-\frac{1}{z+i\sqrt{b}}\right)$.


    From this, I think I am right in saying that if I is the same integral in my previous post but with $\displaystyle b^2$ instead of $\displaystyle b$ then

    $\displaystyle |b| \leq |a|\implies I=0$

    Can someone confirm this for me?
    No, this is the contrary. Or you can say that your former integral is 0 if $\displaystyle |b|>|a|^2$. Indeed, it is zero if the poles $\displaystyle \pm\sqrt{b}$ are outside the circle of integration.
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