Extract the real part of a function as a symbolic expression

I want to extract the real part of a known function. Can anyone point out the reason for which Matlab doesn't want to separate the real and imaginary part of the complex function?

clc, clear
syms w e s m R r I b t z b;

k = i^(3/2)*sqrt(i*w*s*m)*sqrt(1+(i*w*e)/s);
J1 = besselj(1,k*R);
J0 = besselj(0,k*r);

funct = real((k*J0*I*exp(i*(w*t+b*z)))/(2*pi*R*J1));

funct

Answer 1

[Walter Roberson]

It cannot figure out what the real and imaginary parts are for that form. For one thing you have not given any constraints on the variables so it must assume they are all complex valued.

But even if you tell it they are all real, you are calling besselj() with parameters that look like they are probably complex (but perhaps might not be, in some circumstances), and besselj() can generally return real or complex values; certainly MATLAB is not going to be able to predict in detail exactly how changes in the various variables are potentially separable into real and complex parts on the output of besselj.

[Jonathan Velasco]

You might be on to something. I tried to print out just the value of "k", and the same behavior appears. I believe at least up until that point it's related to complex value under the square root. The mistake might come from the syntax in k even before gets to the Bessel functions. I will look into it. Thanks for your input.

[Walter Roberson]

For example Re(k) is

(1/4)*sqrt(2)*sqrt(abs(w*s*m))*(csgn(w*e-I*s)*signum(s)*(signum(w*s*m)-1)*sqrt((signum(s)*sqrt(e^2*w^2+s^2)-s)/s)-(signum(w*s*m)+1)*sqrt((signum(s)*sqrt(e^2*w^2+s^2)+s)/s))

which is some progress but still rather messy. It doesn't get better until you assume all of the variables are nonnegative, in which case you can get (for example) -(1/2)*sqrt(2)*sqrt(w*m*(sqrt(e^2*w^2+s^2)+s))

[Alex Hanes]

For specific cases, you can use assume(variable,'real') from the Symbolic Toolbox to define a variable or parameter as real-valued. This might not work for all cases, particularly when you expect solutions can be complex, but if you want to solve something like the full-width at half-maximum of a Gaussian:

where you know and are real-valued and is half of the maximum. Certainly this equation will give 2 real-valued roots, but you get the basic idea.

% FWHM from symbolic
clear; clc; 

% Define the Gaussian function centered at t0 symbolically
syms y t t0 tau
y = (tau*sqrt(2*pi))^(-1)*exp(-(t-t0)^2/(2*tau^2)); 

% Constrain tau = real and tau > 0 (positive)
assume(tau,'real'); 
assume(tau > 0); 

% Find maximum of function from dy/dt = 0
y_max = solve(diff(y,t) == 0); 

% Find roots of x for y = y_max/2
x_roots = solve(y == y_max/2,t)

FWHM = x_roots(1,1) - x_roots(2,1)

[Walter Roberson]

assume(tau>0) also happens to trigger the assumption of tau being real-valued.

Answer 2

[madhan ravi]

real(funct) %extracts Real part of complex function
imag(funct) %extracts Imaginary part of complex function

[Jonathan Velasco]

That's the expected behavior. However, as an answer I just get the same expression, meaning if my function is f(z), then I get ans = real(f(z)) yet no recombined or simplified version with the real part alone. In other words this is the output:

   ans =

real((2^(1/2)*I*exp(b*z*1i + t*w*1i)*((e*w*1i)/s + 1)^(1/2)*besselj(0, 2^(1/2)*r*((e*w*1i)/s + 1)^(1/2)*(m*s*w*1i)^(1/2)*(- 1/2 + 1i/2))*(m*s*w*1i)^(1/2)*(- 1/4 + 1i/4))/(R*pi*besselj(1, 2^(1/2)*R*((e*w*1i)/s + 1)^(1/2)*(m*s*w*1i)^(1/2)*(- 1/2 + 1i/2)))),

which is the same as me writing the whole expression as I currently have it on paper. Note that the function doesn't split the expression into a + ib

[madhan ravi]

Yes I understand, let me do some research

[Walter Roberson]

Is

(1/8)*I_*2^(1/2)*abs(w*s*m)^(1/2)*Re(BesselJ(0, (1/2)*r*2^(1/2)*abs(w*s*m)^(1/2)*(I*signum(s)*csgn(w*e-I*s)*((signum(s)*(e^2*w^2+s^2)^(1/2)-s)/s)^(1/2)+((signum(s)*(e^2*w^2+s^2)^(1/2)+s)/s)^(1/2)))*(((1+I+(-1+I)*signum(w*s*m))*cos(b*z+t*w)-(1-I+(1+I)*signum(w*s*m))*sin(b*z+t*w))*csgn(w*e-I*s)*signum(s)*((signum(s)*(e^2*w^2+s^2)^(1/2)-s)/s)^(1/2)+((signum(s)*(e^2*w^2+s^2)^(1/2)+s)/s)^(1/2)*((1-I+(1+I)*signum(w*s*m))*cos(b*z+t*w)+(1+I+(-1+I)*signum(w*s*m))*sin(b*z+t*w)))/BesselJ(1, (1/2)*R*2^(1/2)*abs(w*s*m)^(1/2)*(I*signum(s)*csgn(w*e-I*s)*((signum(s)*(e^2*w^2+s^2)^(1/2)-s)/s)^(1/2)+((signum(s)*(e^2*w^2+s^2)^(1/2)+s)/s)^(1/2))))/(Pi*R)

in any way meaningfully more split up for your purposes? In the above, Re() is real part, I_ is your I variable, I is imaginary unit, Pi is pi .

You would have a difficult time getting further than that. You can make a slight bit of progress if you add the constraint that all of your variables are nonnegative, but not much really.

[madhan ravi]

Thank you sir Walter , as he mentioned try adding constraints that is

assume(w,'real') and see further what happens

[madhan ravi]

Likewise for other variables too (adding assumptions for matlab to recognise). Unfortunately I suspect whether you can add imag assumptions to variables as I skimmed through the documentation.

[Walter Roberson]

syms w e s m R r I b t z b real

does not help in MATLAB for this purpose; the expression is too complicated to find the real and imaginary parts of.

Breaking up into real and imaginary parts would require having a decent closed form formula for besselj, which you are not going to find.

[madhan ravi]

Thank you for the clear explanation sir Walter