Welcome to our community

Be a part of something great, join today!

Integration a long closed curve is 0

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
[tex]\int_{\gamma (t) }\, f(z) dz [/tex]

[tex]\int_{\alpha}^{\beta} \, f(\gamma (t))\, \gamma '(t) \, dt [/tex]

[tex]\text{Use the substitution : } \gamma (t) = \xi [/tex]

[tex]\int_{\gamma (\alpha) }^{\gamma (\beta)} \, f(\xi )\, d \xi [/tex]

[tex]\text{If we integrate around a closed loope : }\gamma (\alpha) = \gamma(\beta) [/tex]

[tex]\int_{\gamma (\alpha) }^{\gamma (\alpha)} \, f( \xi )\, d \xi =0 [/tex]

[tex]\text{This is only true if the function is analytic }[/tex]

Feel free to leave any comments .
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,123
[tex]\int_{\gamma (t) }\, f(z) dz [/tex]

[tex]\int_{\alpha}^{\beta} \, f(\gamma (t))\, \gamma '(t) \, dt [/tex]

[tex]\text{Use the substitution : } \gamma (t) = \xi [/tex]

[tex]\int_{\gamma (\alpha) }^{\gamma (\beta)} \, f(\xi )\, d \xi [/tex]

[tex]\text{If we integrate around a closed loope : }\gamma (\alpha) = \gamma(\beta) [/tex]

[tex]\int_{\gamma (\alpha) }^{\gamma (\alpha)} \, f( \xi )\, d \xi =0 [/tex]

[tex]\text{This is only true if the function is analytic }[/tex]

Feel free to leave any comments .
Comment, not a full answer.

This will work if your function f is conservative. I don't how that relates to the analyticity of f.

-Dan
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
Comment, not a full answer.

This will work if your function f is conservative. I don't how that relates to the analyticity of f.

-Dan
But how to define conservative functions mathematically ?
We define analytic functions as those which satisfy the Cauchy-Reimann equations and the partial derivatives exist and are continuous so if they have a pole then we can use the Cauchy-integral formula to find the integral along the loop this is illustrated by the deformation hypothesis .
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,123
But how to define conservative functions mathematically ?
We define analytic functions as those which satisfy the Cauchy-Reimann equations and the partial derivatives exist and are continuous so if they have a pole then we can use the Cauchy-integral formula to find the integral along the loop this is illustrated by the deformation hypothesis .
Ah! It's a complex integration. You didn't tell us that. (Tmi)

Then as far as I know, so long as you have a closed path (that doesn't contain any nasty singularities) then the answer is 0.

-Dan

Come to think about it, if it's analytic I think that means no singularities. I'm too lazy to check that. Time for a nap! (Yawn)

-Dan
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,193
If $f$ is analytic inside the region enclosed by $\gamma$, the integral in question will be zero. Are you putting forth a proof of that? I'm a little unclear what it is you're after.
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
If $f$ is analytic inside the region enclosed by $\gamma$, the integral in question will be zero. Are you putting forth a proof of that? I'm a little unclear what it is you're after.
Yes, indeed.
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,193
What are your assumptions? What theorems are you allowed to use?
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
I am using the contour integration formual :

[tex]\int_{\gamma (t) }\, f(z) dz =\int_{\alpha}^{\beta} \, f(\gamma (t))\, \gamma '(t) \, dt [/tex]