How can I code a contour integral representation of cosine? - matlab

Question

I am pretty new to MATLAB computation and would appreciate any help on this. Firstly, I am trying to integrate a cosine function using the McLaurin expansion [cos(z) = 1 − (z^2)/2 + (z^4)/4 + ....] and lets say plotted over one cycle from 0 to 2π. This first integral (as seen in Figure 1) would serve as a "reference" or "contour" for what I want to do next.
Figure 1 - "The first integral representation"
Now, my next problem comes from writing in MATLAB cos(z) in
terms of an integral in the complex plane.
Figure 2 - "showing cos(z) as an integral in the complex plane"
Where I could choose an equi-sampled set of points 'sn' along the contour, from 'sL' to 'sU'
and separated by '∆s'. This is Euler’s method.
I am trying to write a code to numerically approximate the integral
along a suitable contour and plot the approximation against the
exact value of cos(z). I would like to do this twice - once for
z ∈ [0, 6π] and once for complex valued z in the range
z ∈ [0 + i, 6π + ]. Then to plot both the real and imaginary part of the
computed cos(z)
I am aware of the steps I am looking to implement which I'll bullet-point here.
Choose γ, SL, SU , N.
Step through z from z lower to z upper (use a different
number of steps (other than N) for this).
For each value of z compute cos z by stepping along the contour in
N discrete steps from SL to SU .
For each value of sn along the contour compute the integrand e^(sn-(z^2/4sn))/sqrt(sn) and add it to the rolling sum [I have attached figure 3 showing an image formula of the integrand if its not clear!] Figure 3 - "The exponential integrand I am looking to compute"
Now I will show what I have attempted in MATLAB!
% "Contour Integral Representation of Cosine"
% making 'z' a possible number such as pi
N = 10000000; % example number - meaning sample of steps
z_lower = 0;
z_upper = 6*pi;
%==========================%
z1 = linspace(z_lower,z_upper,N);
y = 1;
Sl = y - 10*1i;
sum = 0.0;
%==========================%
for z = linspace(z_lower,z_upper,N)
for Sn = linspace(Sl,Su,N)
sum = sum + ((exp(Sn) - (z.^2/4*Sn))/sqrt(Sn))*ds;
end
end
sum = sum*(sqrt(pi)/2*pi*1i);
plot(Sn,sum)
Edit1: Hiya, this figure will show what I am expecting - the numerical method to be not exactly the same as the "symbolic" integration lets say. In figure 4, the black cosine wave is the same as in figure 1 and the blue is the numerical integration method.Figure 4 - "End Result of what I expect to plot