alyafey22
Gold Member
MHB
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\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 .
\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 .