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sec5-1

# sec5-1 - ChE 120B THE LAPLACE TRANSFORM 1 series Origin of...

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ChE 120B 5 - 1 T HE L A P LACE T RANSFORM 1. Origin of the LaPlace Transform. A periodic function () f x on the interval x π << can be expanded in a Fourier series 0 1 1 c o s s i n 2 nn n f xa an x b n x = =+ + where 1 cos 1 sin n n af x n x d x bf x n x d x = = Since 11 cos , sin , 22 inx inx inx inx nx e e nx e e i −− = + we can write the Fourier series as n inx n n f xc e =+∞ =−∞ = where we define { 1 for 0 2 ab i n +≥ n c = { { 1 for 0 2 ai b n Then 1 2 inx n cf x e d x = If we wish to represent a function f(x) on an interval x ≥≥ L L as a Fourier series, we simply introduce the new variable x x ′ = L Then we can write inx n f e −∞ ⎛⎞ = ⎜⎟ ⎝⎠ L

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ChE 120B 5 - 2 where 1 2 inx n cf x e d x π ππ ⎛⎞ ′′ = ⎜⎟ ⎝⎠ L and returning to the variable x we have: () / L in x L n fx c e −∞ = (1) where / 1 2 L L in x L n L x e d x L = (2) Thus the function f x defines uniquely the coefficients ( ) L n c on the interval (– L,L ) and vice versa. Now our objective is to expand an aperiodic function in a Fourier series, so we let L →∞ ; in other words we regard the case of an aperiodic function as that of a periodic function with infinite period. If we fix n , /0 nL as L and hence from equation (2) () () / lim 2 lim L L in x L n LL L Lc f x e dx f xdx →∞ →∞ −− == ∫∫ This limit is a single constant and hence cannot determine ( ) f x . We observe however that as L the set of numbers of the form /2 with 0, 1 n , ±2, . .. becomes more and more dense on the real line. This motivates us to replace the quantity / by a continuous variable ω , and to keep ω fixed as L . We obtain the limiting function () () () / ˆˆ lim 2 or ix L L f f ω ωπ ωω →∞ −∞ when L the summation of equation (1) can be replaced by an integral. In doing so, we observe that this is a summation w.r.t n which can be written as ( Δ n = 1): // / / 11 22 L in x L in x L in x L nn n in x L i x ce n L n L n L n e e L πω ∞∞ −∞ −∞ −∞ −∞ −∞ Δ= Δ = Δ ∑∑ Therefore, when L , / can be considered as a continuous variable and the above summation can be substituted by an integral: ˆ 2 L n Lc e d fxe d −∞ −∞ The two relations 1 , 2 f x fx f e d −∞ −∞ (3)
ChE 120B 5 - 3 form a Fourier transform pair. They are equivalent to: () () 1 ˆˆ () , 2 ix f fxe d x fx f e d ωω ω π ∞∞ −− −∞ −∞ == ∫∫ (4) These relations are very useful in dealing with problems for which x is on the real line x −∞ < < ∞ . The trouble is that the Fourier transform of a function f x exists only if the function is absolutely integradable: || fx d x −∞ <∞ (5) Unfortunately, many of the functions in which we shall be interested do not meet this requirement, typical examples being the unit step function and the sine function. It is convenient to modify in the form (consider now functions that are nonzero for 0 x ax x −∞ (6) so that there exists a constant 0 a such that (6) holds. The smallest a for which (6) holds is

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sec5-1 - ChE 120B THE LAPLACE TRANSFORM 1 series Origin of...

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