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
asked Nov 22 at 15:17
Paras Khosla
329
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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
asked Nov 22 at 15:17
Paras Khosla
329
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
add a comment |
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
asked Nov 22 at 15:17
Paras Khosla
329
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
asked Nov 22 at 15:17
Paras Khosla
329
asked Nov 22 at 15:17
Paras Khosla
329
asked Nov 22 at 15:17
Paras Khosla
329
asked Nov 22 at 15:17
Paras Khosla
329
asked Nov 22 at 15:17
Paras Khosla
329
329
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
add a comment |
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
add a comment |
1 Answer
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1
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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
$$
int_gamma pi e^{pi overline{z}}~{rm d}z = -ipi int_0^1 e^{-ipi (1 - t)}{rm d}t = (cdots)
$$
answered Nov 22 at 15:23
caverac
11.5k21027
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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
$$
int_gamma pi e^{pi overline{z}}~{rm d}z = -ipi int_0^1 e^{-ipi (1 - t)}{rm d}t = (cdots)
$$
answered Nov 22 at 15:23
caverac
11.5k21027
add a comment |
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
$$
int_gamma pi e^{pi overline{z}}~{rm d}z = -ipi int_0^1 e^{-ipi (1 - t)}{rm d}t = (cdots)
$$
answered Nov 22 at 15:23
caverac
11.5k21027
add a comment |
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
$$
int_gamma pi e^{pi overline{z}}~{rm d}z = -ipi int_0^1 e^{-ipi (1 - t)}{rm d}t = (cdots)
$$
answered Nov 22 at 15:23
caverac
11.5k21027
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
$$
int_gamma pi e^{pi overline{z}}~{rm d}z = -ipi int_0^1 e^{-ipi (1 - t)}{rm d}t = (cdots)
$$
answered Nov 22 at 15:23
caverac
11.5k21027
answered Nov 22 at 15:23
caverac
11.5k21027
answered Nov 22 at 15:23
caverac
11.5k21027
answered Nov 22 at 15:23
caverac
11.5k21027
11.5k21027
add a comment |
add a comment |
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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