unable to integrate acos(function)

Hi, I am using the symbolic math toolbox to find an expression for the sine of an angle I do have the cosine of, defined as such:

syms theta_Za cos_theta_Zb phi_Za N theta_ab phi_ab theta_b phi_b real

Tay_ord = 5 %truncation order for the Taylor series expansion

A = 3/(8*pi)*sin(theta_Za)*sin(theta_ab)*exp(-1i*(phi_Za-phi_ab));

B = 3/(4*pi)*cos(theta_Za)*cos(theta_ab);

C = 3/(8*pi)*sin(theta_Za)*sin(theta_ab)*exp(1i*(phi_Za-phi_ab));

cos_theta_Zb = ((4*pi)/3)*(A+B+C);

So the most straightforward way of doing this was to include in the expression I need (polX_b) the sine of the arccosine of cos_theta_Zb:

polX_b = (1-exp(-N.*sin(acos(cos_theta_Zb))*cos(phi_b)));

Although, MATLAB does not seem to handle this properly and when I try to include polX_b in the calculation I am interested to perform (Taylor series expansion followed by term by term integration of a product of polX_b and three other (1-exp(...)) expression) I cannot find a closed form solution of the integral. Code shown below:

polZ_a = (1-exp(-N.*cos(theta_Za)));

polZ_b = (1-exp(-N.*(cos_theta_Zb)));

polX_a =(1-exp(-N.*(sin(theta_Za)*cos(phi_Za))));

ZZXX_2_osc = (polZ_a)*(polZ_b)*(polX_a)*(polX_b);

Tay_ZZXX_2_osc = (taylor(ZZXX_2_osc,N,0,'Order',Tay_ord));

int_theta_Tay_ZZXX = simplify((int(Tay_ZZXX_2_osc*sin(theta_Za), theta_Za, [0 pi],'IgnoreAnalyticConstraints',true)))

What could be the most convenient way of fixing this script?

Thanks a lot in advance

Answer 1

[Karan Nandankar]

Hi,

As I can see you are approximating your function 'ZZXX_2_osc' over variable 'N' in taylor expansion, while the integration variable is set as 'theta_Za'. This is probably the reason we are getting an unresolved integral, and int is unable to compute its closed form.

I tried approximating the function over variable 'theta_Za' and that worked for me.

%Approximation using Taylor
tay_ZZXX_2_osc = taylor(ZZXX_2_osc, theta_Za, 'ExpansionPoint', 0, 'Order', Tay_ord);

% Integration
int_theta_Tay_ZZXX = simplify((int(tay_ZZXX_2_osc*sin(theta_Za), theta_Za, [0 pi],'IgnoreAnalyticConstraints',true)))

Hope this helps in your workflow.