I'm currently doing online studies but I'm stuck on this example really need help on how to go about in imageExample 4.6.1 or suggestions on any good book I can use for complex integration.
Example: Find the value of $$\int_C \frac{z+4}{z^2+2z+5} dz$$ where $C$ is the circle $|z + 1| = 1.$
Solution: The poles of the integrand are obtained by setting $z^2 + 2z + 5 = 0,$ from which $z = -1 \pm 2i.$ Since the circle $|z + 1| = 1$ does not enclose any of the two poles, it follows that $$\int_C \frac{z+4}{z^2+2z+5} dz = 0,$$ by the Cauchy Integral Theorem.
The key to this example is that the denominator is never zero on any point interior to the contour you're integrating on. This is because any point interior to or on the contour should satisfy $|z + 1| \leq 1,$ and for our two zeroes $-1 \pm 2i$ we have $|(-1 \pm 2i) + 1| = |2i| = 2 > 1.$
So, by virtue of the quotient rule and the differentiability of polynomials, the integrand is holomorphic (complex-differentiable) everywhere inside the contour. Cauchy's Integral Theorem tells us that if a function is integrated on a closed contour and it is holomorphic over the interior of the contour, then the integral must be zero.
This is essentially a special case of Green's theorem where the Cauchy-Riemann relations for a holomorphic function guarantee that the "curl" must be zero.
Does that help you make sense of this?