Integration a long closed curve is 0

ZaidAlyafey

Well-known member
MHB Math Helper
$$\int_{\gamma (t) }\, f(z) dz$$

$$\int_{\alpha}^{\beta} \, f(\gamma (t))\, \gamma '(t) \, dt$$

$$\text{Use the substitution : } \gamma (t) = \xi$$

$$\int_{\gamma (\alpha) }^{\gamma (\beta)} \, f(\xi )\, d \xi$$

$$\text{If we integrate around a closed loope : }\gamma (\alpha) = \gamma(\beta)$$

$$\int_{\gamma (\alpha) }^{\gamma (\alpha)} \, f( \xi )\, d \xi =0$$

$$\text{This is only true if the function is analytic }$$

Feel free to leave any comments .

topsquark

Well-known member
MHB Math Helper
$$\int_{\gamma (t) }\, f(z) dz$$

$$\int_{\alpha}^{\beta} \, f(\gamma (t))\, \gamma '(t) \, dt$$

$$\text{Use the substitution : } \gamma (t) = \xi$$

$$\int_{\gamma (\alpha) }^{\gamma (\beta)} \, f(\xi )\, d \xi$$

$$\text{If we integrate around a closed loope : }\gamma (\alpha) = \gamma(\beta)$$

$$\int_{\gamma (\alpha) }^{\gamma (\alpha)} \, f( \xi )\, d \xi =0$$

$$\text{This is only true if the function is analytic }$$

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
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
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.

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!

-Dan

Ackbach

Indicium Physicus
Staff member
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
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
What are your assumptions? What theorems are you allowed to use?

ZaidAlyafey

Well-known member
MHB Math Helper
I am using the contour integration formual :

$$\int_{\gamma (t) }\, f(z) dz =\int_{\alpha}^{\beta} \, f(\gamma (t))\, \gamma '(t) \, dt$$