1. ## circle and integral

True or False: $\int_{C} \frac{1}{z^2+1} \ dz = 0$ for every circle given by $|z| = r, \ r > 1$.

So that singular points are at $z^2 = -1$. Thus $z = \pm i$. But these are both inside the circle $|z| = r, \ r > 1$. Thus the statement is false?

2. $\int_{C} \frac{1}{z^2+1} \ dz = 2\pi i(\text{Res}_{z=i} \frac{1}{z^2+1} + \text{Res}_{z=-i} \frac{1}{z^2+1})$

3. Originally Posted by chiph588@
$\int_{C} \frac{1}{z^2+1} \ dz = 2\pi i(\text{Res}_{z=i} \frac{1}{z^2+1} + \text{Res}_{z=-i} \frac{1}{z^2+1})$
So the statement is false?

4. $\text{Res}_{z=i} \frac{1}{z^2+1} = \frac{1}{2i}$

and

$\text{Res}_{z=-i} \frac{1}{z^2+1} = -\frac{1}{2i}$

5. Originally Posted by manjohn12
True or False: $\int_{C} \frac{1}{z^2+1} \ dz = 0$ for every circle given by $|z| = r, \ r > 1$.

So that singular points are at $z^2 = -1$. Thus $z = \pm i$. But these are both inside the circle $|z| = r, \ r > 1$. Thus the statement is false?
This is a special case of a more general result. If $f,g$ are polynomials functions with $g$ non-constant and $\gcd(f,g)=1$ with $\deg (g) \geq \deg (f) + 2$ then the sum of residues of $\tfrac{f}{g}$ is always zero. Try proving it!

6. Originally Posted by ThePerfectHacker
This is a special case of a more general result. If $f,g$ are polynomials functions with $g$ non-constant and $\gcd(f,g)=1$ with $\deg (g) \geq \deg (f) + 2$ then the sum of residues of $\tfrac{f}{g}$ is always zero. Try proving it!
Suppose that the sum of residues of $\tfrac{f}{g}$ was not $0$. Then somehow arrive at a contradiction. Is this something you discovered?

7. Originally Posted by manjohn12
Suppose that the sum of residues of $\tfrac{f}{g}$ was not $0$. Then somehow arrive at a contradiction. Is this something you discovered?
No, I had this problem on a complex analysis exam. I just remembered the result.