Evaluating the Integral of $pi e^{pi overline z}$ with respect to $z$











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I've been given a question as part of the homework on Complex Integration. I cannot seem to think how to integrate it especially with the presence of the conjugate of $z$ in the expression



$$displaystyleint_gamma pi e ^{pi overline z} dz$$ where $gamma$ is the line segment from $i$ to $0$. What method can I use to integrate the expression having $overline z$ with respect to $z$?



Note: Here $overline z$ is the conjugate of $z$






complex-analysis complex-integration







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  • 1




    A straight line from $i$ to $0$? So, the real part will always be $0$ and $overline z$ will be the same as $- z$.
    – badjohn
    Nov 22 at 15:20










  • Would it equal $-2$?
    – Paras Khosla
    Nov 22 at 15:23















up vote
0
down vote

favorite












I've been given a question as part of the homework on Complex Integration. I cannot seem to think how to integrate it especially with the presence of the conjugate of $z$ in the expression



$$displaystyleint_gamma pi e ^{pi overline z} dz$$ where $gamma$ is the line segment from $i$ to $0$. What method can I use to integrate the expression having $overline z$ with respect to $z$?



Note: Here $overline z$ is the conjugate of $z$






complex-analysis complex-integration







share|cite|improve this question


















  • 1




    A straight line from $i$ to $0$? So, the real part will always be $0$ and $overline z$ will be the same as $- z$.
    – badjohn
    Nov 22 at 15:20










  • Would it equal $-2$?
    – Paras Khosla
    Nov 22 at 15:23













up vote
0
down vote

favorite









up vote
0
down vote

favorite











I've been given a question as part of the homework on Complex Integration. I cannot seem to think how to integrate it especially with the presence of the conjugate of $z$ in the expression



$$displaystyleint_gamma pi e ^{pi overline z} dz$$ where $gamma$ is the line segment from $i$ to $0$. What method can I use to integrate the expression having $overline z$ with respect to $z$?



Note: Here $overline z$ is the conjugate of $z$






complex-analysis complex-integration







share|cite|improve this question













I've been given a question as part of the homework on Complex Integration. I cannot seem to think how to integrate it especially with the presence of the conjugate of $z$ in the expression



$$displaystyleint_gamma pi e ^{pi overline z} dz$$ where $gamma$ is the line segment from $i$ to $0$. What method can I use to integrate the expression having $overline z$ with respect to $z$?



Note: Here $overline z$ is the conjugate of $z$






complex-analysis complex-integration




complex-analysis complex-integration






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 22 at 15:17









Paras Khosla

449




449








  • 1




    A straight line from $i$ to $0$? So, the real part will always be $0$ and $overline z$ will be the same as $- z$.
    – badjohn
    Nov 22 at 15:20










  • Would it equal $-2$?
    – Paras Khosla
    Nov 22 at 15:23














  • 1




    A straight line from $i$ to $0$? So, the real part will always be $0$ and $overline z$ will be the same as $- z$.
    – badjohn
    Nov 22 at 15:20










  • Would it equal $-2$?
    – Paras Khosla
    Nov 22 at 15:23








1




1




A straight line from $i$ to $0$? So, the real part will always be $0$ and $overline z$ will be the same as $- z$.
– badjohn
Nov 22 at 15:20




A straight line from $i$ to $0$? So, the real part will always be $0$ and $overline z$ will be the same as $- z$.
– badjohn
Nov 22 at 15:20












Would it equal $-2$?
– Paras Khosla
Nov 22 at 15:23




Would it equal $-2$?
– Paras Khosla
Nov 22 at 15:23










1 Answer
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Parametrize the path $gamma$:



$$
gamma = {z = x + i y ~|~ x = 0 ~mbox{ and } y = 1 - t, ~ 0leq 0 leq 1}
$$



Along this path



$$
{rm d}z = -i{rm d}t
$$



So that



$$
int_gamma pi e^{pi overline{z}}~{rm d}z = -ipi int_0^1 e^{-ipi (1 - t)}{rm d}t = (cdots)
$$






share|cite|improve this answer





















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    Parametrize the path $gamma$:



    $$
    gamma = {z = x + i y ~|~ x = 0 ~mbox{ and } y = 1 - t, ~ 0leq 0 leq 1}
    $$



    Along this path



    $$
    {rm d}z = -i{rm d}t
    $$



    So that



    $$
    int_gamma pi e^{pi overline{z}}~{rm d}z = -ipi int_0^1 e^{-ipi (1 - t)}{rm d}t = (cdots)
    $$






    share|cite|improve this answer

























      up vote
      1
      down vote













      Parametrize the path $gamma$:



      $$
      gamma = {z = x + i y ~|~ x = 0 ~mbox{ and } y = 1 - t, ~ 0leq 0 leq 1}
      $$



      Along this path



      $$
      {rm d}z = -i{rm d}t
      $$



      So that



      $$
      int_gamma pi e^{pi overline{z}}~{rm d}z = -ipi int_0^1 e^{-ipi (1 - t)}{rm d}t = (cdots)
      $$






      share|cite|improve this answer























        up vote
        1
        down vote










        up vote
        1
        down vote









        Parametrize the path $gamma$:



        $$
        gamma = {z = x + i y ~|~ x = 0 ~mbox{ and } y = 1 - t, ~ 0leq 0 leq 1}
        $$



        Along this path



        $$
        {rm d}z = -i{rm d}t
        $$



        So that



        $$
        int_gamma pi e^{pi overline{z}}~{rm d}z = -ipi int_0^1 e^{-ipi (1 - t)}{rm d}t = (cdots)
        $$






        share|cite|improve this answer












        Parametrize the path $gamma$:



        $$
        gamma = {z = x + i y ~|~ x = 0 ~mbox{ and } y = 1 - t, ~ 0leq 0 leq 1}
        $$



        Along this path



        $$
        {rm d}z = -i{rm d}t
        $$



        So that



        $$
        int_gamma pi e^{pi overline{z}}~{rm d}z = -ipi int_0^1 e^{-ipi (1 - t)}{rm d}t = (cdots)
        $$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 22 at 15:23









        caverac

        12.8k21028




        12.8k21028






























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