Question

The value of integral e^2z÷z-2 dz where |z| = 3 is

Answer 1

Question :

Evaluate $\sf\oint_{c}\:\dfrac{e^{2z}\:dz}{(z-2)}$,where |z|= 3

Solution :

Let f(z) = e²ᶻ

Now, $\sf\oint_{c}\:\dfrac{e^{2z}\:dz}{(z-2)}$

$\sf\oint_{c}\:\dfrac{f(z)\:dz}{(z-2)}$

Since e²ᶻ is Analytic inside the circle,C : |z|=3 and a=2

Therefore, By Cauchy's integral formula :

$\sf\oint_{c}\:\dfrac{f(z)\:dz}{(z-2)}=2\pi\:i\times\:f(a)$

$\sf=2\pi\:i(e^{2(a)})$

$\sf=2\pi\:i(e^{2(2)})$

$\sf=2\pi\:i\:e^4$

Therefore, The value of $\sf\oint_{c}\:\dfrac{e^{2z}\:dz}{(z-2)}$ is 2πie⁴.

Theory :

Cauchy's integral formula :

If f(z) is Analytic within and on a closed curve and if a is any point within C, then

$\bf\oint_{c}\:\dfrac{f(z)\:dz}{(z-a)}=2\pi\:i\:f(a)$