Fourier Transform of Power Series

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Global JournalJournal of Science Frontier Research: F Mathematics and Decision Sciences Volume 18 Issue 7 VersionVersion 1.0 Year 2018 Type: Double Blind Peer Reviewed International Research Journal PPublisublisher: Global Journals Online ISSN: 2249-4626 & Print ISSN: 0975-5896

Fourier Transform of Power Series By Shiferaw Geremew Kebede, Awel Seid Gelete, Dereje Legesse Abaire & Mekonnen Gudeta Gizaw MadMaddda Walabu University Abstract- The authors establish a set of presumably new results, which provide Fourier transform of power series. So in this paper the author try to evaluate Fourier transform of some challenging functions by expressing them as a sum of infinitely terms. Hence, the method is useful to find the Fourier transform of functions that difficult to obtain their Fourier transform by ordinary method or using definition of Fourier transformations. Keywords: fourier transforms, power series, taylor's and maclaurin series and gamma function. GJSFR-F Classification: FORFOR Code: MSC 2010: 35S30

FourierTransformofPowerSeries

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Fourier Transform of Power Series

Notes Shiferaw Geremew Kebede α, Awel Seid Gelete σ, Dereje Legesse Abaire ρ & Mekonnen Gudeta Gizaw Ѡ 201

Abstract- The authors establish a set of presumably new results, which provide Fourier transform of power series. So in r ea this paper the author try to evaluate Fourier transform of some challenging functions by expressing them as a sum of Y infinitely terms. Hence, the method is useful to find the Fourier transform of functions that difficult to obtain their Fourier transform by ordinary method or using definition of Fourier transformations. 491 Keywords: fourier transforms, power series, taylor's and maclaurin series and gamma function. I. Introduction V The Fourier transform is one of the most important integral transforms. Be-cause of a number of special properties, it is very useful in studying linear differential equations. VII ue ersion I s Fourier analysis has its most important applications in mathematical modeling, s

physical and engineering and solving partial differential equations (PDEs) related to I boundary and initial value problems of Mechanics, heat flow, electro statistics and other fields. Daniel Bernoulli (1700-1782) and Leonhard Euler (1707-1783), Swiss XVIII mathematicians, and Jean-Baptiste D Alembert (1717-1783), a French mathematician, physicist, philosopher, and music theorist, were all prominent in the ensuing mathematical music debate. In 1751, Bernoullis memoir of 1741-1743 took Rameaus findings into

)

account, and in 1752, DAlembert published Elements of theoretical and practical music F according to the principals of Monsieur Rameau, clarified, developed, and simplified. (

DAlembert was also led to a differential equation from Taylors problem of the vibrating string, ∂2y ∂2y Research Volume = α2 ∂x2 ∂2t2

The current widespread use of the transform (mainly in engineering) came about Frontier during and soon after World War II, although it had been used in the 19th century by Abel, Lerch, Heaviside, and Bromwich. Science Joseph Fourier's method of Fourier series for solving the diffusion equation could only apply to a limited region of space because those solutions were periodic. In 1809, of Laplace applied his transform to find solutions that diffused indefinitely in space.

a) Definition Journal The Fourier transform of the function f(x) is given by: Global 1 Z ∞ F (f(x)) = √ f(x)e−ixωdx 2Π 0

b) Definition A Power series is a series deﬁned of the form:

Author: Department of Mathematics, Madda Walabu University, Bale Robe Ethiopia. e-mail: [email protected]

Fourier Transform of Power Series

∞ n f(x) = Σn=0an(x − c)

2 3 n = a0 + a1(x − c) + a2(x − c) + a3(x − c) + ... + an(x − c) where c is any constant c ∈ <

c) Definition If f(x) has a power series expansion at c,where c is any constant c ∈ <. It’s Taylor’s series expansion is: Notes (x − c)n 201 ∞ (n) f(x) = Σn=0anf (c) r n!

ea Y d) Definition 1 50 Maclaurin Series expansion of the function f(x) is: xn f(x) = Σ∞ f (n)(0) n=0 n! V

00 000 VII 0 f (0) f (0) = f(0) + f (0)x + x2 + x3 + ... ue ersion I

s 2! 3! s I e) Definition

XVIII The gamma function, whose symbol Γ(s) is deﬁned when s > 0 by the formula Z ∞ −x n−1

) Γ = e x dx

F 0 (

II. Fourier Transform of Power Series

Research Volume Theorem 1: (Fourier Transform of power series) Iff(x) has a Power series expansion at c,where c is any constant c ∈ <.

∞ n Frontier f(x) = Σn=0an(x − c) then the Fourier transform of f(x) is given in the form of power series as: Science

of ∞ n F (f(x)) = F (Σn=0an(x − c) )

1 ∞ 1 Γ(n + 1) Journal = √ Σ an 2Πeicω n=0 (iω)n+1 sn+1

Global Proof Suppose f(x) has a Power series expansion at c,where c is any constant c ∈ <. i.e

∞ n f(x) = Σn=0an(x − c)

Then, By using the deﬁnition of Fourier transforms,

Fourier Transform of Power Series

F (f(x)) = F (Σ∞ a (x − c)n) n=0 n Z ∞ 1 ∞ n −ixω = √ [Σn=0an(x − c) ]e dx 2Π −∞ Z ∞ 1 ∞ n −ixω = √ Σn=0 an(x − c) e dx 2Π −∞ Z ∞ otes 1 ∞ −ixω n N = √ Σn=0 ane (x − c) dx 2Π −∞

1 Z ∞ 201 √ ∞ −ixω n = Σ0 an e (x − c) dx r −∞ 2Π ea Y Let,x = t + c ⇐⇒ dx = dt 511 So, ∞ n F (f(x)) = F (Σn=0an(x − c) ) V Z ∞ 1 ∞ −i(t+c)ω n VII = √ Σ0 an e t dt

2Π −∞ ue ersion I s s

Z ∞ I 1 ∞ −itω −icω n = √ Σ0 an e e t dt 2Π −∞ XVIII

Z ∞ 1 ∞ −icω −itω n = √ Σ0 ane e t dt

−∞ 2Π ) F ( Let, v = itω ⇐⇒ t = v ⇒ dt = 1 dv iω iω Hence,

Z ∞ Research Volume 1 ∞ −icω −v v n 1 F (f(x)) = √ Σ0 ane e [ ] dv 2Π −∞ iω iω

Z ∞ n Frontier 1 ∞ −icω −v v 1 = √ Σ0 ane e dv 2Π −∞ (iω)n iω

Z ∞ Science 1 ∞ 1 −icω −v n = √ Σ0 an e e v dv of 2Π [iω]n+1 −∞ Z ∞ 1 ∞ 1 −icω −v n = √ Σ0 an e [2 e v dv] Journal 2Π [iω]n+1 0

1 ∞ 1 −icω Γ(n + 1) Global = √ Σ an e [2 ] 2Π 0 [iω]n+1 sn+1

2 ∞ 1 Γ(n + 1) = √ Σ an 2Πeicω 0 [iω]n+1 sn+1 In particular, for n = 1, 2, 3, ... Γ(n + 1) = n!

Fourier Transform of Power Series

Such that, ∞ n 2 ∞ 1 n! F (Σ an(x − c) ) = √ Σ an n=0 2Πeicω 0 [iω]n+1 sn+1 Theorem 2:) (Fourier Transform of Taylor’s Series) If f(x) has a power series expansion at c,where c is any constant c ∈ <. It’s Taylor’s series expansion is: otes (x − c)n N f(x) = Σ∞ a f (n)(c) n=0 n n! 201

r then the Fourier transform of f(x) is given in the form of power series as:

ea Y (x − c)n F (f(x)) = F [Σ∞ a f (n)(c) ] 521 n=0 n n!

2 (n) ∞ 1 Γ(n + 1) = √ f (c)Σ an 2Πeicω n=0 n!(iω)n+1 sn+1 V VII Proof ue ersion I s s Suppose f(x) has a Power series expansion at c,where c is any constant I c ∈ <.

XVIII Hence, the Taylor’s series expansion of f(x) is:

n ∞ (n) (x − c) f(x) = Σn=0anf (c)

n! ) F ( Then, By using the deﬁnition of Fourier transforms,

(x − c)n F (f(x)) = F (Σ∞ a f (n)(c) ) Research Volume n=0 n n! 1 Z ∞ (x − c)n = √ [Σ∞ a f (n)(c) ]e−ixωdx Frontier n=0 n 2Π −∞ n!

Z ∞ n Science 1 ∞ (n) (x − c) −ixω = √ Σn=0anf (c) e dx

of 2Π −∞ n!

1 Z ∞ 1 = √ Σ∞ a f (n)(c) e−ixω(x − c)ndx Journal n=0 n 2Π −∞ n! 1 1 Z ∞

Global ∞ (n) −ixω n = √ Σ0 anf (c) e (x − c) dx 2Π n! −∞

Let,x = t + c ⇐⇒ dx = dt So, n ∞ (n) (x − c) F (f(x)) = F (Σn=0anf (c) n!

Fourier Transform of Power Series

Z ∞ 1 ∞ (n) 1 −i(t+c)ω n = √ Σ0 anf (c) e t dt 2Π n! −∞ Z ∞ 1 ∞ (n) 1 −itω −icω n = √ Σ0 anf (c ) e e t dt 2Π n! −∞ Z ∞ 1 ∞ (n) 1 −icω −itω n = √ Σ0 anf (c) e e t dt 2Π n! −∞ otes v 1 N Let, v = itω ⇐⇒ t = iω ⇒ dt = iω dv Hence, 1 1 Z ∞ v 1 201

∞ (n) −icω −v n r F (f(x)) = √ Σ anf (c) e e [ ] dv

0 ea −∞ 2Π n! iω iω Y

Z ∞ n 531 1 ∞ (n) 1 −icω −v v 1 = √ Σ0 anf (c) e e dv 2Π n! −∞ (iω)n iω Z ∞ 1 ∞ (n) 1 1 −icω −v n = √ Σ0 anf (c) e e v dv V 2Π n! [iω]n+1 −∞ VII Z ∞ 1 1 1 ue ersion I

∞ (n) −icω −v n s = √ Σ0 anf (c) e [2 e v dv] s n! [iω]n+1 0 2Π I

1 ∞ (n) 1 1 −icω Γ(n + 1) XVIII = √ Σ anf (c) e [2 ] 2Π 0 n! [iω]n+1 sn+1 2 1 1 Γ(n + 1)

∞ (n) = √ Σ0 anf (c) ) icω n! [iω]n+1 sn+1 F

2Πe (

In particular, for n = 1, 2, 3, ...

Research Volume Γ(n + 1) = n! Such that, Frontier n ∞ (n) (x − c) 2 ∞ (n) 1 1 n! F (Σ anf (c) ) = √ Σ anf (c) n=0 n! icω 0 n! [iω]n+1 sn+1

2Πe Science

Theorem 3:) (Fourier Transform of Maclaurin Series) of In particular if f(x) has a power series expansion at 0, then, the power

series expansion of f(x) is given by: Journal

∞ n f(x) = Σn=0anx Global which is known as Maclaurin series, then the Fourier transform of f(x) is deﬁned by:

∞ n F (f(x)) = F (Σn=0anx )

2 (n) ∞ 1 Γ(n + 1) = √ f (c)Σ an 2Π n=0 (iω)n+1 sn+1

Fourier Transform of Power Series

proof

suppose f(x) has the power series expansion at 0 i.e ∞ n f(x) = Σn=0anx By using the deﬁnition of Fourier transforms, F (f(x)) = F (Σ∞ a xn) n=0 n Notes Z ∞ 1 ∞ n −ixω = √ [Σ anx ]e dx

201 n=0 2Π −∞ r

ea Z ∞ Y 1 ∞ n −ixω = √ Σn=0anx e dx 541 2Π −∞ Z ∞ 1 ∞ −ixω n = √ Σn=0 ane x dx 2Π −∞ V Z ∞ 1 ∞ −ixω n VII = √ Σ0 an e x dx 2Π −∞ ue ersion I s s t 1 I Let, t = ixω ⇐⇒ x = iω ⇒ dx = iω dt Hence,

XVIII Z ∞ 1 ∞ −t t n 1 F (f(x)) = √ Σ0 an e [ ] dt 2Π −∞ iω iω

) 1 Z ∞ tn 1 F ∞ −t

( = √ Σ0 an e dt 2Π −∞ (iω)n iω

Z ∞ 1 ∞ 1 −t n = √ Σ0 an e t dt Research Volume 2Π [iω]n+1 −∞ 1 1 Z ∞ √ ∞ −t n = Σ0 an n+1 [2 e t dt] Frontier 2Π [iω] 0

1 ∞ 1 Γ(n + 1) = √ Σ0 an [2 ] Science 2Π [iω]n+1 sn+1 of 2 ∞ 1 Γ(n + 1) = √ Σ an 2Π 0 [iω]n+1 sn+1 Journal Note: In particular, for n = 1, 2, 3,

Global Γ(n + 1) = n! Hence,

∞ n 2 ∞ 1 n! F (Σ anx ) = √ Σ an n=0 2Π 0 [iω]n+1 sn+1

Fourier Transform of Power Series

III. Conclusion The results on Fourier transform of power series are summarized as follows; 2 Some functions like et , sint and son on are difficult to get their Fourier transform. t Hence it is possible to find Fourier transform such functions by expanding them into power series, Taylor's series and Maclaurin series form as: 2 ∞ 1 Γ(n + 1) F (f(x)) = √ Σ an icω 0 n+1 n+1 Notes 2Πe [iω] s where ∞ n f(x) = Σn=0an(x − c) , c ∈ < 201 r

2. ea Y 2 ∞ (n) 1 1 Γ(n + 1) F (f(x)) = √ Σ anf (c) 2Πeicω 0 n! [iω]n+1 sn+1 551 where (x − c)n f(x) = Σ∞ a f (n)(c) n=0 n n! V 3. VII 2 ∞ 1 Γ(n + 1)

F (f(x)) = √ Σ0 an ue ersion I n+1 n+1 s 2Π [iω ] s s where I ∞ n f(x) = Σn=0ant XVIII

References Références Referencias

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1. Shiferaw Geremew Kebede, Properties of Fourier cosine and sine transforms, Basic and F Applied Research IJSBAR, 35(3) (2017) 184-193 (

2. Shiferaw Geremew Kebede, Properties of Fourier Cosine and Sine Integrals with the Product of Power and Polynomial Functions, International Journal of Sciences: Basic

and Applied Research (IJSBAR) (2017) Volume 36, No 7, pp 1-17 Research Volume 3. Janelle k. Hammond, UW-L Journal of undergraduate Research XIV(2011) 4. Ordinary Differential equations, GABRIEL NAGY, Mathematics Department, SEPTEMBER 14, 2015 5. Advanced Engineering Mathematics, Erwin Kreyszing, Herbert Kreyszing,Edward J. Frontier Norminton 10th Edition 6. A first Course in Di_erential equations, Rudolph E. Longer, 1954 7. Differential Equation and Integral Equations, Peter J. Collins, 2006 Science 8. Differential Equations, James R. Brannan, William E. Boyce, 2nd edition of 9. Differential Equations for Engineers, Wei-Chau Xie, 2010 10. Advanced Engineering Mathematics 7th Edition, PETER V. ONEIL 11. Historically, how and why was the Laplace Transform invented? Written 18 Oct 2015 Journal From Wikipedia: Global