Here a beginner in complex variable analysis. They ask me to evaluate the following integral using the Cauchy integral formula or the Cauchy integral theorem
$$\frac{1}{2 \pi i} \oint \frac{(\cos z+\operatorname{sen} z)}{\left(z^{2}+25\right)(z+1)} d z \quad \text { about } \frac{x^{2}}{9}+\frac{y^{2}}{16}=1$$
I do not understand very well how to use it and it would be very helpful if you could recommend bibliography
This is the Cauchy integral formula:
Theorem. Let $f$ analytic everywhere within and on a simple closed contour $C$, taken in the positive sense. if $z_{0}$ is any ponint interior to $C$, then
$$ f(z_{0}) = \frac{1}{2\pi i} \oint_{C} \frac{f(z)}{z-z_{0}} dz$$
The reference is Churchill et. al. Complex variables and applications.
In your case
$$ C= \left\{z = x+iy\in \mathbb{C} \big| \frac{x^{2}}{9}+\frac{y^{2}}{16}=1 \right\}$$
Note that $C$ in an ellipse in the complex plane with vertex at $x= \pm 3$ and covertex at $y=\pm 4$
The integrand $\displaystyle \frac{(\cos z+\operatorname{sen} z)}{\left(z^{2}+25\right)(z+1)}$ has three singularities:
$$ z = \pm i5 \Longrightarrow y = \pm 5 $$
and
$$ z = -1 \Longrightarrow x = -1$$
So, the singularities at $y=\pm i5$ are out of the covertex range, the only singularity inside $C$ is $z=-1$
Then, if you define
$$f(z) = \frac{(\cos z+\operatorname{sen} z)}{\left(z^{2}+25\right)}$$
the function $f$ is analytic within and on the contour $C$, the ellipse. If you put $z_{0} = -1$ you have all the ingredients to apply the theorem.