# How to deduce Fourier / Z transform from Laplace transform through a CAS-Calculator.

I have a calculator with CAS ( https://en.wikipedia.org/wiki/Computer_algebra_system ) and in particular I have a Casio Algebra FX 2.0 Plus ( https://en.wikipedia.org/wiki/List_of_computer_algebra_systems ).

I need to compute Laplace / Zeta / Fourier Transform.

My calculator is able to calculate laplace transforms, in fact if I go in the CAS section and insert:

"$$∫ (sinX * e-SX, X)$$"

it gives me as a result:

" $$-e^(-s·x)·(COS (x) + s · SIN (x)) / (s ^ 2 + 1)$$"

then evaluating: $$LIM (- e ^(-sx)·(COS (x) + s · SIN (x)) / (s ^ 2 + 1), x, ∞) - LIM (- e^(-s·x)·(COS (x) + s · SIN (x)) / (s ^ 2 + 1), x, 0 ) = 1 / (s ^ 2 + 1)$$.

And indeed

$$ℒ_x [sin (x) * step (x)] (s) = 1 / (s ^ 2 + 1)$$.

I tried to do it even with more complex expressions, and I always get the correct result.

Unfortunately, I can not find a way to calculate the Z-Transform or the Fourier Transform, using the CAS of my calculator .

At this point I wondered if you knew:

OR 1) A method to calculate also Z-transform and Fourier-Transform by using the CAS of my calculator.

OR 2) Calculated the laplace transform of a function (since, as proven, my calculator is able to do this), then easily deduce from it the Z-Transform and the Fourier-Transform.

OR 3) Since my calculator also accepts C / C ++ programs, do you know any program in this language for the calculation of Fourier Transform / Zeta Transform? Thanks everyone in advance for the help.

NOTES:

Laplace Transform: https://en.wikipedia.org/wiki/Laplace_transform

Z-Transfrom: https://en.wikipedia.org/wiki/Z-transform

Fourier Trnasform: https://en.wikipedia.org/wiki/Fourier_transform

I do not have this calculator, but according to the manual, you can compute generalized integrals (it supports definite integration and provides the infinity symbol). Thus, you can compute the Laplace transform in a single and simpler command using the definition. If the Fourier transform is defined (or equivalently if the imaginary axis is included in the ROC for the Laplace transform), you can use the substitution $$s = j \omega$$ to find the Fourier Transform.
Now, Z-transform applies to discrete signals, so there is no direct connection. The only connection I know between the Laplace transform and the Z-transform is that the second appears when you ideally sample a function, take its Laplace transform and set $$z = e^{sT}$$, where T is the sampling period (see Starred Transform). However, this will not assist you in finding the Z-transform efficiently. Nevertheless, you should be able to use the "sum" function in your calculator to attempt to find the Z-transform directly from its definition.
• As far as your second question is concerned, notice that I told you that you can use $s=j \omega$ if the Fourier transform is defined (equivalently if the imaginary axis is included in the Lapalce transform's ROC). Note that this is not the case for $sin(x)$ since the definition integral diverges. That is why, in order to talk about its "Fourier Transfrom" you have to extend the context of the original definition to include generalized functions. But for these, the substitution $s = j \omega$ no longer works. Jan 18 '19 at 21:59