MHB Why is the integral of $\frac{1}{z+2}$ over $\gamma(0;1)$ equal to zero?

[shen07]

Hello i would like to know why this integral is Zero:

$$\int_{\gamma(0;1)}\frac{1}{z+2} \mathrm{d}z$$

Well i know by a fundamental result that:
$$\int_{\gamma(a;r)}\frac{1}{z-a} \mathrm{d}z=2\pi\imath$$

But here the point $$z=-2$$ lies outside $$\gamma(0;1)$$ so what is the reasoning behind could you help me please?

 

[I like Serena]

Re: Reason why Integral Zero

Hi shen07! :)

But here the point $$z=-2$$ lies outside $$\gamma(0;1)$$ so what is the reasoning behind could you help me please?

I believe you already said it: the "pole" lies outside the closed curve.
In that case the circular integral is zero.
If the curve were around a pole, it would be $2\pi i$.

 

[Prove It]

Re: Reason why Integral Zero

Hello i would like to know why this integral is Zero:

$$\int_{\gamma(0;1)}\frac{1}{z+2} \mathrm{d}z$$

Well i know by a fundamental result that:
$$\int_{\gamma(a;r)}\frac{1}{z-a} \mathrm{d}z=2\pi\imath$$

But here the point $$z=-2$$ lies outside $$\gamma(0;1)$$ so what is the reasoning behind could you help me please?

Cauchy's Integral Theorem states that for any complex function which is closed and holomorphic everywhere in and on the boundary, its contour integral is equal to 0. As your function only has a singular point at z = -2, which is not in the boundary, your function satisfies the conditions and so Cauchy's Integral Theorem applies.

Proof of the Theorem

 

[alyafey22]

Re: Reason why Integral Zero

Well i know by a fundamental result that:
$$\int_{\gamma(a;r)}\frac{1}{z-a} \mathrm{d}z=2\pi\imath$$

By using $$\gamma(a;r)$$ you are making sure that the pole at $$z=a$$ is included in your curve by letting it the center of the disk since the radius is nonzero .

 

[shen07]

Re: Reason why Integral Zero

What if i use Cauchy's Theorem, since $$f(z)=\frac{1}{z+2}$$ is holomorphic on and inside $$\gamma(0;1)$$, using Cauchy's Theorem
$$\int_{\gamma(0;1)}\frac{1}{z+2} \mathrm{d}z=0$$

 

[alyafey22]

Re: Reason why Integral Zero

What if i use Cauchy's Theorem, since $$f(z)=\frac{1}{z+2}$$ is holomorphic on and inside $$\gamma(0;1)$$, using Cauchy's Theorem
$$\int_{\gamma(0;1)}\frac{1}{z+2} \mathrm{d}z=0$$

Yes , since the only pole is at $z=-2$ which is out of the circular curve .