Complex line integrals
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Suppose we have an analytic function then Why complex integral of that function does not depend on the path of integration?
complex-analysis contour-integration line-integrals analytic-functions
asked Nov 19 at 12:31
robin
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Suppose we have an analytic function then Why complex integral of that function does not depend on the path of integration?
complex-analysis contour-integration line-integrals analytic-functions
asked Nov 19 at 12:31
robin
183
Read about Cauchy's theorem.
– Richard Martin
Nov 19 at 12:34
Because the integral around a closed loop is zero. This follows from Green's theorem in the plane. Look in any book on complex variables.
– saulspatz
Nov 19 at 12:35
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up vote
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down vote
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Suppose we have an analytic function then Why complex integral of that function does not depend on the path of integration?
complex-analysis contour-integration line-integrals analytic-functions
asked Nov 19 at 12:31
robin
183
Suppose we have an analytic function then Why complex integral of that function does not depend on the path of integration?
complex-analysis contour-integration line-integrals analytic-functions
complex-analysis contour-integration line-integrals analytic-functions
asked Nov 19 at 12:31
robin
183
asked Nov 19 at 12:31
robin
183
asked Nov 19 at 12:31
robin
183
asked Nov 19 at 12:31
robin
183
asked Nov 19 at 12:31
robin
183
183
Read about Cauchy's theorem.
– Richard Martin
Nov 19 at 12:34
Because the integral around a closed loop is zero. This follows from Green's theorem in the plane. Look in any book on complex variables.
– saulspatz
Nov 19 at 12:35
add a comment |
Read about Cauchy's theorem.
– Richard Martin
Nov 19 at 12:34
Because the integral around a closed loop is zero. This follows from Green's theorem in the plane. Look in any book on complex variables.
– saulspatz
Nov 19 at 12:35
Read about Cauchy's theorem.
– Richard Martin
Nov 19 at 12:34
Read about Cauchy's theorem.
– Richard Martin
Nov 19 at 12:34
Because the integral around a closed loop is zero. This follows from Green's theorem in the plane. Look in any book on complex variables.
– saulspatz
Nov 19 at 12:35
Because the integral around a closed loop is zero. This follows from Green's theorem in the plane. Look in any book on complex variables.
– saulspatz
Nov 19 at 12:35
add a comment |
1 Answer
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In general, the complex integral depends on the path of integration !
Example: $D= mathbb C setminus {0}, f(z)=1/z$ and $c(t)=e^{it}$ with $t in [0, 2m pi i]$ for some $m in mathbb N$.
Then we have $int_c f(z) dz = 2m pi i$.
answered Nov 19 at 12:36
Fred
43.5k1644
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
In general, the complex integral depends on the path of integration !
Example: $D= mathbb C setminus {0}, f(z)=1/z$ and $c(t)=e^{it}$ with $t in [0, 2m pi i]$ for some $m in mathbb N$.
Then we have $int_c f(z) dz = 2m pi i$.
answered Nov 19 at 12:36
Fred
43.5k1644
add a comment |
up vote
1
down vote
In general, the complex integral depends on the path of integration !
Example: $D= mathbb C setminus {0}, f(z)=1/z$ and $c(t)=e^{it}$ with $t in [0, 2m pi i]$ for some $m in mathbb N$.
Then we have $int_c f(z) dz = 2m pi i$.
answered Nov 19 at 12:36
Fred
43.5k1644
add a comment |
up vote
1
down vote
up vote
1
down vote
In general, the complex integral depends on the path of integration !
Example: $D= mathbb C setminus {0}, f(z)=1/z$ and $c(t)=e^{it}$ with $t in [0, 2m pi i]$ for some $m in mathbb N$.
Then we have $int_c f(z) dz = 2m pi i$.
answered Nov 19 at 12:36
Fred
43.5k1644
In general, the complex integral depends on the path of integration !
Example: $D= mathbb C setminus {0}, f(z)=1/z$ and $c(t)=e^{it}$ with $t in [0, 2m pi i]$ for some $m in mathbb N$.
Then we have $int_c f(z) dz = 2m pi i$.
answered Nov 19 at 12:36
Fred
43.5k1644
answered Nov 19 at 12:36
Fred
43.5k1644
answered Nov 19 at 12:36
Fred
43.5k1644
answered Nov 19 at 12:36
Fred
43.5k1644
43.5k1644
add a comment |
add a comment |
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Read about Cauchy's theorem.
– Richard Martin
Nov 19 at 12:34
Because the integral around a closed loop is zero. This follows from Green's theorem in the plane. Look in any book on complex variables.
– saulspatz
Nov 19 at 12:35