Laplace Series (PDF)

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					                            Laplace Series
Laplace Series

Laplacetransformation is used in the mathematics, if we require any change in the
value then transformation is used to check the function. Laplace transformation is
mainly used in the integral and differential equations. When we solve the starting
value then we use Fourier Series and Laplace Series. When we increase the limit of
integration then we get two sided Laplace transformation. It is also used to find the
transient system or frequency range of a system.

Now we see laplaceseries.

The spherical harmonic function form a complete orthogonal system, and we have an
arbitrary real function f ( ∅, z) ≡ ∑i=0 ∑m = 0 tends to ∞.

So we can write it as:

f (∅, z) ≡ ∑i=0 ∑m = 0 Plm Qlm (∅, z),

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Or in term of real spherical harmonic:

So we can write this series as:

f (∅, z) ≡ ∑i=0 ∑m = 0 [Rlm Qlmc (∅, z) + Slm Qlmcs (∅, z)]

And the representation of a function f (∅, z) is such as Fourier series also said to be
Laplace series.

Now see the process of determining the coefficient Plm, i.e. multiply both sides by
Qlm’ and we use orthogonally relationship to obtain Laplace series.

⇒∫02? ∫0? f (∅, z) Qlm’ (∅, z) sin ∅d∅;
⇒∑i=0 ∑m = 0 ∫02? ∫0? Plm Qlm’ (∅, z) Qlm’ sin ∅d∅;
⇒∑i=0 ∑m = 0 Plm
⇒ Plm

Now we see some of the formulas of Laplace transformation:
The formula of Laplace transformation is given below:

1. L[pn] = n!
           Rn + 1;
Inverse of Laplace is:
L-1 [ 1 ] = 1 Pn - 1
    Rn (n – 1)!;

2.L[eap] = 1

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           R– a;
Inverse of above Laplace transform is:
L-1 [ 1 ] = eap;
  (R – a)

3. L[sin ap] = a
             R2 + a2;
And the inverse of above transform is:

L-1 [ 1 ] = 1sin ap;
(R2 + a2) a

4. L[cos ap] = R
             R2 + a2;
And the inverse of above Laplace is:

L-1 [ R ] = cos ap ;
(R2 + a2)                       Page No. : ­ 3/4
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