What is the proper way to apply absolute value on cauchy's formula? Given Cauchy's formula for calculating the value of a complex function using the integral around it:
$$f\left(z\right)=\frac{1}{2{\pi}i}\int_\Gamma\frac{f\left(s\right)}{s-z}ds$$
What is the proper way to apply absolute value on it, what we did is:
$$\left|f\left(z\right)\right|\le\frac{1}{2{\pi}}\int_\Gamma\frac{\left|f(s)\right|}{\left|s-z\right|}ds$$
We are not sure about whether it should be $=$ or $\le$, and also to make sure that this is the correct way to apply the absolute value of the integral.
 A: There is a problem because $s$ is a complex variable. You need write it as a integral over a real parameter, then you can apply the absolute value with the $\le$ sign.
Example that it doesn't work before that: $f(z)=1$, $z=0$, $\Gamma = \{s:|s|=1\}$
$$\left|\frac{1}{2\pi i}\oint_{|s|=1}\frac{1}{s}ds\right| = |\frac{1}{2\pi i} 2\pi i| = 1$$
$$\frac{1}{2\pi} \oint_{|s|=1}\frac{1}{|s|}ds = \frac{1}{2\pi} \oint_{|s|=1} ds = 0$$
In a way, you need to apply absolute value to $ds$ as well.
A: Yes, your application of the absolute value is correct. The following image may give you some intuition as to why it needs to be $\leq$ rather than $=$. Note that in the left integral, the area below the $x$-axis cancels some of the area below the $x$-axis and therefore
$$
\left|\int f(x) dx \right| \leq \int |f(x)| dx .
$$
Even for complex numbers $x$, on the left they can cancel, while on the right they cannot.

The mathematical proof of this fact follows from the triangle inequality for the norm:
$$
|x_1+x_2| \leq |x_1| + |x_2|
$$
By induction you get
$$
\left| \sum_{i=1}^n x_i \right| \leq \sum_{i=1}^n |x_i|
$$
and (Rieman)-integrals are defined as limits of sums and therefore
\begin{align}
\left| \int_{a}^{b} f(x) dx \right| &= \left| \lim_{n \to\infty}  \sum_{i=0}^{n-1} f\left(a+i\cdot \frac{b-a}{n}\right)\cdot \frac{b-a}{n} \right| \\
&\leq  \lim_{n \to\infty} \sum_{i=0}^{n-1}  \left|  f\left(a+i\cdot \frac{b-a}{n}\right)  \right|  \cdot \frac{b-a}{n} = \int_a^b |f(x)| dx .
\end{align}
