MATLAB: Symbolic expansions, is it possible to express a solution as a series of exponentials

collectexpandexponentialsymbolic

Lets say I have something like

f_t = (A*cos(a*t) + B*sin(b*t))*(C*cos(c*t) + D*sin(c*t)),

what I would like is something of the form

f_t = c_1 exp(k1*t) + c_2 exp(k2*t) + … + c_N exp(kN*t) .

I can't find anything within the symbolic toolbox that allows expansion in this form. I don't really want to write one either until I know for sure there isn't one included / on the FEX. Anyone that can help?

Best Answer

There is one in the Symbolic Math Toolbox: rewrite!

To wit:

syms A a t
f_t = (A*cos(a*t) + B*sin(b*t))*(C*cos(c*t) + D*sin(c*t));
f_t = rewrite(f_t, 'exp')

produces:

f_t =
(A*(exp(-a*t*i)/2 + exp(a*t*i)/2) + B*((exp(-b*t*i)*i)/2 - (exp(b*t*i)*i)/2))*(C*(exp(-c*t*i)/2 + exp(c*t*i)/2) + D*((exp(-c*t*i)*i)/2 - (exp(c*t*i)*i)/2))

You’ll have to play with it it get the result in your example. There may not be any direct way to do that.

Related Solutions

MATLAB: How to remove some specific terms from a function

For that h and m, and holding a and t symbolic, but assuming a is real and assuming t is positive:

    YourIntegral = 284475670516978489457516559826613190 * exp(-284475670516978489457516559826613190 * x^2 / (i*t + 568951341033956978915033119653226380 * a^2)) * 2^(1/2) * pi^(1/2)/ ((142237835258489244728758279913306595*i) * t + 80926407116084504595296678347317302904777180422918746308915785881976100 * a^2)^(1/2)

To get to this form instead of a piecewise form, before you do the integral use assume() to add the assumptions about a and t

assume(t > 0 & a > 0)

and you might need to simplify() the result of the integral.

By examination we can see that the x variable exists as a multiplier of the numerator of the exp() term, and does not otherwise appear anywhere in the integral. So to get rid of the exp() term,

exp_removed = subs(YourIntegral, x, sym(0));

Now if you want you could turn it into a numeric procedure with a and t as parameters:

f = matlabFunction( simplify(exp_removed), 'vars', {a, t});

Note: the result is going to be complex.

Also note: there is no factorial or gamma function involved.

MATLAB: Inverse Laplace transform – there has to be a better way

Try this:

syms s t 
num = 9;
denom = [1 9 9 0];
% [r,p,k] = residue(num,denom);
nums = num;
dens = poly2sym(denom, s);                              % Create Symbolic Polynomial
F = nums / dens * 1/s                                   % Transfer Function With Step Input
Fpf = partfrac(F)                                       % Partial Fraction Expansion
f = ilaplace(Fpf)                                       % Time Domain Expression
f = rewrite(f, 'exp')                                   % Rewrite As Exponential Terms
f = vpa(f,4)                                            % Convert Fractions To Decimal Equivalents
flatex = latex(f)                                       % LaTeX Expression

producing:

Best Answer

Related Solutions

MATLAB: How to remove some specific terms from a function

MATLAB: Inverse Laplace transform – there has to be a better way

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