# How can I code a contour integral representation of cosine?

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% "Contour Integral Representation of Cosine"

%==========================%

% making 'z' a possible number such as pi

N = 10000000; % example number

z_lower = 0;

z_upper = 6*pi;

%==========================%

z1 = linspace(z_lower,z_upper,N);

y = 1;

Sl = y - 10*1i;

Su = y + 10*1i;

ds = (Su - Sl)/N; % size of each step

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)

The above is my poor attempt! 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 + i]. Then I intend to plot both the real and imaginary part of the computed cos z.

I have a following lost of steps that I am trying to implement!

- Choose γ, Sl, Su , N.
- Step through z from z lower to z upper (using 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 "((exp(Sn) - (z.^2/4*Sn))/sqrt(Sn))" [I have attached an image to make this integrand clearer to read] and add it to the rolling sum.
- Plot the result.
- Repeat using a better integration rule

Image of the integrand!

I am pretty new to general forms of computation and would really appreciate any help! Thanks!

##### 8 Comments

### Answers (1)

David Goodmanson
on 24 Oct 2022

Edited: David Goodmanson
on 24 Oct 2022

Hi Richard,

this integral converges very slowly along the vertical path gamma-i*inf to gamma+i*inf.

Convergence depends on what happens for large values of s. In that case, you can drop the z^2/s term in the exponential and look at exp(s). When the path of integration has large imaginary part, exp(s) is oscillatory and only 1/sqrt(s) is making the integrand smaller. Convergence is slow.

Rather than stick to the vertical path, which will prove the point philosophically but is difficult to achieve numerically, it helps to use a different one. The path is moved over from the original path in a continuous fashion. Here the new path is a rectangle:

-inf+ia a+ib

------<-------

o ^ o = origin

------>------^

-inf-ia a-ib

The idea is that at the left hand parts of the path, s has a large negative real part and exp(s) dies off quickly rather than oscillating.

In the code below it would have been nice to use the set

I1 = integral(@(t) fun(t,z), -inf-i*a, b-i*a);

I2 = integral(@(t) fun(t,z), b-i*a, b+i*a);

I3 = integral(@(t) fun(t,z), b+i*a, -inf+i*a);

but the integral function can't handle complex end points at infinity, so the vertical offset is inserted manually in I1 and I3. With the choices for a and b the code does well up to z = 7*pi. You can mess around with a and b to see what works.

z = pi/3;

a = 3

b = 1

I1 = integral(@(t) fun(t,z,-i*a), -inf, b);

I2 = integral(@(t) fun(t,z,0 ), b-i*a, b+i*a);

I3 = integral(@(t) fun(t,z, i*a), b, -inf);

I = sqrt(pi)/(2*pi*i)*(I1+I2+I3)

delta = I - cos(z) % check

function y = fun(t,z,u)

s = t+u;

y = (exp(s-z.^2./(4*s))./sqrt(s));

end

I = 0.5000 + 0.0000i

delta = -2.7756e-16 + 6.2638e-17i

##### 2 Comments

David Goodmanson
on 24 Oct 2022

Edited: David Goodmanson
on 25 Oct 2022

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