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Thread: Contour integration

  1. #1
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    Contour integration

    given intergrate from 0 to 2pi

    (sin t)^2 dz / jz
    ---------
    5- 4 cos t

    i achieve the C_-1 equation till

    ( z^2 + z^-2 - 2) ( -j )
    ----------------------- dz
    -20z + 8z^2 + 8

    The pole are 2 & 0.5. ( Only 0.5 is lying in |z|=1 )

    But i can't get the final answer which is ( pi/4) when i sub z =0.5 into

    ( z^2 + z^-2 - 2) * (2pi)
    ------------------
    -----20+16z



    many thanks
    ck
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  2. #2
    Senior Member DeMath's Avatar
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    Quote Originally Posted by Chris0724 View Post
    given intergrate from 0 to 2pi

    (sin t)^2 dz / jz
    ---------
    5- 4 cos t

    i achieve the C_-1 equation till

    ( z^2 + z^-2 - 2) ( -j )
    ----------------------- dz
    -20z + 8z^2 + 8

    The pole are 2 & 0.5. ( Only 0.5 is lying in |z|=1 )

    But i can't get the final answer which is ( pi/4) when i sub z =0.5 into

    ( z^2 + z^-2 - 2) * (2pi)
    ------------------
    -----20+16z
    many thanks
    ck
    $\displaystyle \int\limits_0^{2\pi } {\frac{{{{\sin }^2}t}}
    {{5 - 4\cos t}}\,dt} = \left\{ \begin{gathered}
    {e^{it}} = z, \hfill \\
    dt = \frac{{dz}}
    {{iz}} \hfill \\
    \end{gathered} \right\} = \frac{1}
    {{4i}}\oint\limits_{\left| z \right| = 1} {\frac{{{z^4} - 2{z^2} + 1}}
    {{{z^2}\left( {2{z^2} - 5z + 2} \right)}}\,dz} .$

    $\displaystyle f\left( z \right) = \frac{{{z^4} - 2{z^2} + 1}}{{{z^2}\left( {2z - 1} \right)\left( {z - 2} \right)}}.$

    So, you have three poles: $\displaystyle {z_{1,2}} = 0,{\text{ }}{z_3} = \frac{1}{2},{\text{ }}{z_4} = 2$. Note that the pole $\displaystyle {z_{1,2}} = 0 $ is the double and $\displaystyle {z_4} = 2$ isn't lying in the unit circle $\displaystyle |z|=1$.
    Then, according to the residue theorem and Jordan's lemma, you have

    $\displaystyle \frac{1}{{4i}}\oint\limits_{\left| z \right| = 1} {\frac{{{z^4} - 2{z^2} + 1}}{{{z^2}\left( {2{z^2} - 5z + 2} \right)}}\,dz} = \frac{1}{{4i}} \cdot 2\pi i\left[ {\mathop {{\text{Res}}}\limits_{z \to 0} \bigr\{ {f\left( z \right)} \bigr\} + \mathop {{\text{Res}}}\limits_{z \to \frac{1}{2}} \bigr\{ {f\left( z \right)} \bigr\}} \right].$

    $\displaystyle {\color{red}\boxed{\color{black}\begin{gathered}
    \mathop {{\text{Res}}}\limits_{z \to a} \bigr\{ {f\left( z \right)} \bigr\} = \frac{1}
    {{\left( {n - 1} \right)!}}\mathop {\lim }\limits_{z \to a} \frac{d^{n-1}}
    {{d{z^{n - 1}}}}\Bigl[\left(z - a\right)^n f\left( z \right) \Bigr] \hfill \\
    a{\text{ is the pole point and }}n{\text{ is the pole multiplicity}} \hfill \\
    \end{gathered}}}$

    $\displaystyle \mathop {{\text{Res}}}\limits_{z \to 0} \bigr\{ {f\left( z \right)} \bigr\} = \frac{1}
    {{\left( {2 - 1} \right)!}}\mathop {\lim }\limits_{z \to 0} \frac{{{d^{2 - 1}}}}
    {{d{z^{2 - 1}}}}\left[ {{{\left( {z - 0} \right)}^2}\frac{{{z^4} - 2{z^2} + 1}}
    {{{z^2}\left( {2{z^2} - 5z + 2} \right)}}} \right] =$

    $\displaystyle = \mathop {\lim }\limits_{z \to 0} \frac{d}
    {{dz}}\left[ {\frac{{{z^4} - 2{z^2} + 1}}
    {{2{z^2} - 5z + 2}}} \right] = \mathop {\lim }\limits_{z \to 0} \frac{{4{z^5} - 15{z^4} + 10{z^2} - 12z + 8{z^3} + 5}}
    {{4{z^4} - 20{z^3} + 33{z^2} - 20z + 4}} = \frac{5}{4}.$

    $\displaystyle \mathop {{\text{Res}}}\limits_{z \to \frac{1}{2}} \bigr\{ {f\left( z \right)} \bigr\} = \mathop {\lim }\limits_{z \to \frac{1}{2}} \left( {z - \frac{1}{2}} \right)\frac{{{z^4} - 2{z^2} + 1}}{{{z^2}\left( {2z - 1} \right) \left( {z - 2} \right)}} = \frac{1}{2}\mathop {\lim }\limits_{z \to \frac{1}{2}} \frac{{{z^4} - 2{z^2} + 1}}{{{z^3} - 2{z^2}}} = - \frac{3}
    {4}.$

    Finally, you have

    $\displaystyle \int\limits_0^{2\pi } {\frac{{{{\sin }^2}t}}{{5 - 4\cos t}}\,dt} = \frac{1}{{4i}}\oint\limits_{\left| z \right| = 1} {\frac{{{z^4} - 2{z^2} + 1}}{{{z^2}\left( {2{z^2} - 5z + 2} \right)}}\,dz} = \frac{1}{{4i}} \cdot 2\pi i\left( {\frac{5}{4} - \frac{3}{4}} \right) = \frac{\pi }{4}.$
    Last edited by DeMath; Sep 29th 2009 at 10:18 PM.
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  3. #3
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    Quote Originally Posted by DeMath View Post
    $\displaystyle \int\limits_0^{2\pi } {\frac{{{{\sin }^2}t}}
    {{5 - 4\cos t}}\,dt} = \left\{ \begin{gathered}
    {e^{it}} = z, \hfill \\
    dt = \frac{{dz}}
    {{iz}} \hfill \\
    \end{gathered} \right\} = \frac{1}
    {{4i}}\oint\limits_{\left| z \right| = 1} {\frac{{{z^4} - 2{z^2} + 1}}
    {{{z^2}\left( {2{z^2} - 5z + 2} \right)}}\,dz} .$

    $\displaystyle f\left( z \right) = \frac{{{z^4} - 2{z^2} + 1}}{{{z^2}\left( {2z - 1} \right)\left( {z - 2} \right)}}.$

    So, you have three poles: $\displaystyle {z_{1,2}} = 0,{\text{ }}{z_3} = \frac{1}{2},{\text{ }}{z_4} = 2$. Note that the pole $\displaystyle {z_{1,2}} = 0 $ is the double and $\displaystyle {z_4} = 2$ isn't lying in the unit circle $\displaystyle |z|=1$.
    Then, according to the residue theorem and Jordan's lemma, you have

    $\displaystyle \frac{1}{{4i}}\oint\limits_{\left| z \right| = 1} {\frac{{{z^4} - 2{z^2} + 1}}{{{z^2}\left( {2{z^2} - 5z + 2} \right)}}\,dz} = \frac{1}{{4i}} \cdot 2\pi i\left[ {\mathop {{\text{Res}}}\limits_{z \to 0} \bigr\{ {f\left( z \right)} \bigr\} + \mathop {{\text{Res}}}\limits_{z \to \frac{1}{2}} \bigr\{ {f\left( z \right)} \bigr\}} \right].$

    $\displaystyle {\color{red}\boxed{\color{black}\begin{gathered}
    \mathop {{\text{Res}}}\limits_{z \to a} \bigr\{ {f\left( z \right)} \bigr\} = \frac{1}
    {{\left( {n - 1} \right)!}}\mathop {\lim }\limits_{z \to a} \frac{d^{n-1}}
    {{d{z^{n - 1}}}}\Bigl[\left(z - a\right)^n f\left( z \right) \Bigr] \hfill \\
    a{\text{ is the pole point and }}n{\text{ is the pole multiplicity}} \hfill \\
    \end{gathered}}}$

    $\displaystyle \mathop {{\text{Res}}}\limits_{z \to 0} \bigr\{ {f\left( z \right)} \bigr\} = \frac{1}
    {{\left( {2 - 1} \right)!}}\mathop {\lim }\limits_{z \to 0} \frac{{{d^{2 - 1}}}}
    {{d{z^{2 - 1}}}}\left[ {{{\left( {z - 0} \right)}^2}\frac{{{z^4} - 2{z^2} + 1}}
    {{{z^2}\left( {2{z^2} - 5z + 2} \right)}}} \right] =$

    $\displaystyle = \mathop {\lim }\limits_{z \to 0} \frac{d}
    {{dz}}\left[ {\frac{{{z^4} - 2{z^2} + 1}}
    {{2{z^2} - 5z + 2}}} \right] = \mathop {\lim }\limits_{z \to 0} \frac{{4{z^5} - 15{z^4} + 10{z^2} - 12z + 8{z^3} + 5}}
    {{4{z^4} - 20{z^3} + 33{z^2} - 20z + 4}} = \frac{5}{4}.$

    $\displaystyle \mathop {{\text{Res}}}\limits_{z \to \frac{1}{2}} \bigr\{ {f\left( z \right)} \bigr\} = \mathop {\lim }\limits_{z \to \frac{1}{2}} \left( {z - \frac{1}{2}} \right)\frac{{{z^4} - 2{z^2} + 1}}{{{z^2}\left( {2z - 1} \right) \left( {z - 2} \right)}} = \frac{1}{2}\mathop {\lim }\limits_{z \to \frac{1}{2}} \frac{{{z^4} - 2{z^2} + 1}}{{{z^3} - 2{z^2}}} = - \frac{3}
    {4}.$

    Finally, you have

    $\displaystyle \int\limits_0^{2\pi } {\frac{{{{\sin }^2}t}}{{5 - 4\cos t}}\,dt} = \frac{1}{{4i}}\oint\limits_{\left| z \right| = 1} {\frac{{{z^4} - 2{z^2} + 1}}{{{z^2}\left( {2{z^2} - 5z + 2} \right)}}\,dz} = \frac{1}{{4i}} \cdot 2\pi i\left( {\frac{5}{4} - \frac{3}{4}} \right) = \frac{\pi }{4}.$

    hi DeMath,

    Many thanks for the help

    i dun understand how you get 3 pole... why is it a must to multiply both numerator & denominator by z^2 ?
    Last edited by Chris0724; Sep 30th 2009 at 01:19 AM. Reason: wrong letter
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  4. #4
    Senior Member DeMath's Avatar
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    Quote Originally Posted by Chris0724 View Post
    hi DeMath,

    Many thanks for the help

    i dun understand how you get 3 pole... why is it a must to multiply both numerator & denominator by z^2 ?
    This is just the usual transformation:

    $\displaystyle \int\limits_0^{2\pi } {\frac{{{{\sin }^2}t}}{{5 - 4\cos t}}dt}.$

    Numerator: $\displaystyle {\sin ^2}t = {\left( {\frac{{{e^{it}} - {e^{ - it}}}}
    {{2i}}} \right)^2} = - \frac{1}
    {4}\left( {{e^{2it}} - 2 + \frac{1}
    {{{e^{2it}}}}} \right) = - \frac{1}
    {{4{e^{2it}}}}\left( {{e^{4it}} - 2{e^{2it}} + 1} \right).$

    Denominator: $\displaystyle 5 - 4\cos t = 5 - 2\left( {{e^{it}} + {e^{ - it}}} \right) = - \frac{1}{{{e^{it}}}}\left( {2{e^{2it}} - 5{e^{it}} + 2} \right).$

    Together: $\displaystyle \frac{{{{\sin }^2}t}}
    {{5 - 4\cos t}} = \frac{{ - \frac{1}
    {{4{e^{2it}}}}\left( {{e^{4it}} - 2{e^{2it}} + 1} \right)}}
    {{ - \frac{1}
    {{{e^{it}}}}\left( {2{e^{2it}} - 5{e^{it}} + 2} \right)}} = \frac{1}
    {4}\frac{{{e^{4it}} - 2{e^{2it}} + 1}}
    {{{e^{it}}\left( {2{e^{2it}} - 5{e^{it}} + 2} \right)}}.$

    Then

    $\displaystyle \int\limits_0^{2\pi } {\frac{{{{\sin }^2}t}}
    {{5 - 4\cos t}}\,dt} = \frac{1}
    {4}\int\limits_0^{2\pi } {\frac{{{e^{4it}} - 2{e^{2it}} + 1}}
    {{{e^{it}}\left( {2{e^{2it}} - 5{e^{it}} + 2} \right)}}\,dt} = \left\{ \begin{gathered}
    {e^{it}} = z, \hfill \\
    dt = \frac{{dz}}
    {{iz}} \hfill \\
    \end{gathered} \right\} =$

    $\displaystyle = \frac{1}{4}\oint\limits_{\left| z \right| = 1} {\frac{{{z^4} - 2{z^2} + 1}}
    {{z\left( {2{z^2} - 5z + 2} \right)}}\frac{{dz}}{{iz}}} = \frac{1}{{4i}} \oint\limits_{\left| z \right| = 1} {\frac{{{z^4} - 2{z^2} + 1}}{{{z^2} \left( {2{z^2} - 5z + 2} \right)}}\,dz} .$
    Last edited by DeMath; Sep 30th 2009 at 06:04 AM.
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