"Laplace Series (PDF)"
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), Know More About Derivatives Of Hyperbolic Trig Functions Tutorcircle.com Page No. : 1/4 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 Read More About Derivative Of Fractions Tutorcircle.com Page No. : 2/4 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) Tutorcircle.com Page No. : 3/4 Page No. : 2/3 Thank You For Watching Presentation