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Giving Laplace transform a second look

Posted: 31 Mar 2015     Print Version  Bookmark and Share

Keywords:CERN  Laplace transform  Fourier transform 

The Laplace transform is an integral transform that is arguably second only to the Fourier transform in terms of application in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits. In all its complexity, the Laplace transform offers an alternative functional description that often simplifies the process of analysing the behaviour of the system.

Can you find the error(s), if any, in the following mathematical derivation that applies the Laplace transform? Begin by expressing the function to be transformed, g(t), in the form of the square of f(t), or g(t) = f 2(t). Then by the definition of the Laplace transform,

Laplace transform

Apply the integration by parts formula for definite integrals:

Laplace transform

where u and v are both functions of t, and ti and tf are the integration limits of t. Choose

Laplace transform

and

Laplace transform

Combining these in the formula,

Laplace transform

where the integration limits that applied to t are converted to f. To convert them, apply the condition on f (t) that

Laplace transform

Thus,

Laplace transform

The integration is with respect to f, not t, and consequently the limits of integration have been changed to f (0) and f (∞), as indicated. When the integral is taken and the resulting expression is simplified,

Laplace transform

Therefore, the Laplace transform of all functions of the form g(t) are G(s) = 0.

Nobody who has worked with Laplace transforms would accept this result as valid. Where is the error?

Hint: one error is in the line that reads

Laplace transform

To find v = v(t), the indefinite integral of the integrand f(t)?e?st must be taken, then the limits of integration applied to it. The error is in equating expressions that are not necessarily equal:

Laplace transform

If v were equal to F(s), then v(t) is constant with respect to F(s) and dv/dt = 0, which is consistent with the faulty conclusion.

Erik Margan, sensor-electronics designer of the ATLAS experiment of the Large Hadron Collider (LHC) at CERN found additional errors. Can you identify them?

- Dennis Feucht
  EDN





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