Issue with Laplace Transform for Exponential Functions in MATLAB

Hello everyone,

I'm encountering an issue while trying to compute the Laplace transform of exponential functions in MATLAB using the laplace command. Specifically, when I input an exponential function directly (e.g., f = 2^(-t)), the command does not return a meaningful result. However, if I rewrite the exponential function using the exp function (e.g., exp(-log(2)*t)), the Laplace transform works as expected.

Could someone explain why this discrepancy occurs? Is there a specific syntax or method I should follow to ensure that MATLAB computes the Laplace transform of exponential functions directly, without needing to use exp?

Interestingly, if f = exp(1)^(-t), the laplace command does not work either.

syms t real
f1 = 2^(-t);
f2 = exp(-log(2)*t);
f3 = exp(1)^(-t);
f4 = exp(-t);

F1 = laplace(f1)

F1 = 

F2 = laplace(f2)

F2 = 

F3 = laplace(f3)

F3 = 

F4 = laplace(f4)

F4 = 

I've found some discussion online about this issue, but it wasn't particularly helpful. Here's the link to what I found: MathWorks Discussion.

Thanks in advance for your help!

Answer 1

[Paul]

Hi FFU,

Let's take a look at these one by one.

syms t real
f1 = 2^(-t);

I've never seen a Laplace transform table that includes an explicit entry for this form of a function, but I'd like to. Maybe that's why laplace fails?

F1 = laplace(f1)

F1 = 

But we can do

F1 = laplace(rewrite(f1,'exp'))

F1 = 

f2 = exp(-log(2)*t);

f2(t) is just a form for exp(-a*t), which has transform F2(s) = 1/(s + a), which laplace handles easily.

laplace(f2)

ans = 

Note that log(2) is being approximated in floating point and then expressed as a rational number on conversion to sym. It might be more clear to say

f2 = exp(-log(sym(2))*t)

f2 = 

then

F2 = laplace(f2)

F2 = 

Turning to f3

f3 = exp(1)^(-t)

f3 = 

Again, exp(1) is being evaluated in floating point with numeric exp, so not exactly equal to e, and then converted to a sym rational expression. So now we are back to the same situation as for f1, i.e., a function of the form a^(-t) (with a ~= e)

laplace(f3)

ans = 

laplace(rewrite(f3,'exp'))

ans = 

Putting sym on the inside uses the symbolic function exp (which, amazingly, doesn't have its own doc page I don't believe)

f3 = exp(sym(1))^(-t)

f3 = 

which works fine

laplace(f3)

ans = 

and is, of course, the same as f4

f4 = exp(-t);
laplace(f4)

ans = 

[FFU]

The way the exponential function is handled feels somewhat cumbersome, especially when compared to other symbolic math software.What do you think?

[Paul]

If by "exponential function" you mean functions of the form of f1, I do find it curious I can't find such a function in Laplace trasnsform tables. That is, why don't the tables include an entry for the general case with whatever limitations may apply to a and b (should there be any)?

syms t real
syms a b
f1 = a^(b*t);
laplace(rewrite(f1,'exp'))

ans = 

Maybe it's just expected that one can rewrite f1 in terms of exp so that there's no need for an additional table entry.

I'm not familar with any other symbolic math software, so can't comment on that. If you think Matlab should handle f1 better, you can always submit an Enhancement Request to Tech Support.