Plotting infinite series with set of parameters

I want to plot the following functions,

Here, and given by

is a set of values of values given by,

I have approximated a set of values for . I have tried the following code, but failed to generate expected plot. Is that due to approximation, or there is any other error in my code?

clc
clear all
syms Y
k = 0.1;
j=[-8.0671,-5.30732,-2.62768,0,8.0671,5.30732,2.62768];
for i=1:length(j)
  f1 = (sin(j(i))+ k*j(i)*cos(j(i)))*cos(j(i)*Y) - (cos(j(i)) - k*j(i)*sin(j(i)))*sin(j(i)*Y);
  f2=((((j(i).^4)/16)+1)*((k*(k + 1)*sin(j(i))) - (1 + 2*k - k^.2*j(i).^2)*((cos(j(i)))/(2*j(i)))));
  U=0.25*sum(((f1/f2)*exp(-0.25*j(i).^2)),i,1:8);    
  fplot(U);xlim([0 1]);%hold on;
end

[darova]

Maybe it's a bit complicated for symbolic function fplot()?

Did you try to plot numerically?

What about . Where is τ? Is it 2D plot (surface)?

[Moslem Uddin]

I have considered $\tau=1$

[Moslem Uddin]

It’s not a surface plot.

Answer 1

[darova]

Looks like simple sin() or cos():

What do you think?

Try my script, i think i reached a success

[Moslem Uddin]

Thanks! your code provides solution very close to my expectation. Maybe, the discrepancy is due to approximation. Is there any way to find all values of ? is given by,

[darova]

Try this

k = 0.1;
f = @(j) tan(j) + 2*k*j./(1-k^2*j.^2);
j0 = zeros(1,20);       % find 20 roots
j0(1) = fsolve(f,0);    % find first root
for i = 1:length(j0)-1
    j0(i+1) = j0(i) + 3.5;  % initial guess for next root
    j0(i+1) = fsolve(f,j0(i+1));
end
x = linspace(0,60,500);
y = f(x);
ix = abs(y)>10;         % find big numbers
y(ix) = nan;            % replace big numbers with nan (better plot)
plot(x,y)
hold on
plot(j0,f(j0),'.r')     % plot roots
hold off

    % looks like straight line
figure
plot(j0,'.-')

Maybe some verification will be needed. For k = 0.5 not all roots are good:

EDIT: edited function

[Moslem Uddin]

How to find list of roots instead of plotting and utilise them for the above plot?

[darova]

How many roots you want to find? There are infinite of them

Why not utilise?

ix = abs(f(j0))>0.01;            % find if value bigger than 0.01
if sum(ix)
    j1 = j0(~ix);                % utilise
    plot(j1,f(j1),'.-r')
end

[Moslem Uddin]

I want find roots between -100 to 100 and want to use them in the following code given by you what to do?,

clc,clear

k = 0.1;
R = 4;
R0 = 1;
tau = 1;
y = linspace(0,2,50);
j = [-8.0671, -5.30732, -2.62768, 0, 8.0671, 5.30732, 2.62768];
[Y,J] = ndgrid(y,j);
sj = sin(J);                    % shortcuts
cj = cos(J);

F1 = (sj + k*J.*cj).*cos(J.*Y) - (cj - k*J.*sj).*sin(J.*Y);
F2 = (J.^4/R^2+1).* (k*(k+1).*sj - (1+2*k-J.^2*k^2).*cj/2./J);
E = R0/R * exp(-J.^2*tau/R);
U = sum(F1./F2.*E, 2);           % sum along 2d dimension (j)

plot(y,U)
xlabel('Y AXIS')
ylabel('U AXIS')

[darova]

Read my comments in the code, they should be helpfull

And accept my answer please

[Moslem Uddin]

Thanks!

[darova]

Do you have a source formula? Hard to see it in the script

[darova]

Try to write clearer script. Hard to understand it if everything is written in one line

You forgot the dot

Also declaring y two times

[Moslem Uddin]

Thanks a lot!