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 Strictly as per the compliance and regulations of: © 2018. Shiferaw Geremew Kebede, Awel Seid Gelete, Dereje Legesse Abaire & Mekonnen Gudeta Gizaw. This is a resresearcearch/review paper, distributed under the terms of the Creative Commons Attribution-NoncoNoncommmercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0nc/3.0/),/), permitting all non cocommmercial use, distribution, and reproduction in any mmededium, provided the original work is properly cited. Fourier Transform of Power Series Notes Shiferaw Geremew Kebede α, Awel Seid Gelete σ, Dereje Legesse Abaire ρ & Mekonnen Gudeta Gizaw Ѡ 2018 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 1 F (f(x)) = p f(x)e−ix!dx 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] ©2018 Global Journals Fourier Transform of Power Series 1 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 2 < c) Definition If f(x) has a power series expansion at c,where c is any constant c 2 <. It's Taylor's series expansion is: Notes (x − c)n 2018 1 (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) = Σ1 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 1 −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 2 <. 1 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 1 n F (f(x)) = F (Σn=0an(x − c) ) 1 1 1 Γ(n + 1) Journal = p Σ an 2Πeic! n=0 (i!)n+1 sn+1 Global Proof Suppose f(x) has a Power series expansion at c,where c is any constant c 2 <. i.e 1 n f(x) = Σn=0an(x − c) Then, By using the definition of Fourier transforms, ©2018 Global Journals Fourier Transform of Power Series 1 n F (f(x)) = F (Σn=0an(x − c) ) Z 1 1 1 n −ix! = p [Σn=0an(x − c) ]e dx 2Π −∞ Z 1 1 1 n −ix! = p Σn=0 an(x − c) e dx 2Π −∞ Z 1 otes 1 1 −ix! n N = p Σn=0 ane (x − c) dx 2Π −∞ 1 Z 1 2018 p 1 −ix! n = Σ0 an e (x − c) dx r −∞ 2Π ea Y Let,x = t + c () dx = dt 511 So, 1 n F (f(x)) = F (Σn=0an(x − c) ) V Z 1 1 1 −i(t+c)! n VII = p Σ0 an e t dt 2Π −∞ ue ersion I s s Z 1 I 1 1 −it! −ic! n = p Σ0 an e e t dt 2Π −∞ XVIII Z 1 1 1 −ic! −it! n = p Σ0 ane e t dt −∞ 2Π ) F ( Let, v = it! () t = v ) dt = 1 dv i! i! Hence, Z 1 Research Volume 1 1 −ic! −v v n 1 F (f(x)) = p Σ0 ane e [ ] dv 2Π −∞ i! i! Z 1 n Frontier 1 1 −ic! −v v 1 = p Σ0 ane e dv 2Π −∞ (i!)n i! Z 1 Science 1 1 1 −ic! −v n = p Σ0 an e e v dv of 2Π [i!]n+1 −∞ Z 1 1 1 1 −ic! −v n = p Σ0 an e [2 e v dv] Journal 2Π [i!]n+1 0 1 1 1 −ic! Γ(n + 1) Global = p Σ an e [2 ] 2Π 0 [i!]n+1 sn+1 2 1 1 Γ(n + 1) = p Σ an 2Πeic! 0 [i!]n+1 sn+1 In particular, for n = 1; 2; 3; ::: Γ(n + 1) = n! ©2018 Global Journals Fourier Transform of Power Series Such that, 1 n 2 1 1 n! F (Σ an(x − c) ) = p Σ an n=0 2Πeic! 0 [i!]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 2 <. It's Taylor's series expansion is: otes (x − c)n N f(x) = Σ1 a f (n)(c) n=0 n n! 2018 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 [Σ1 a f (n)(c) ] 521 n=0 n n! 2 1 Γ(n + 1) (n) 1 = p f (c)Σ an 2Πeic! n=0 n!(i!)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 2 <. XVIII Hence, the Taylor's series expansion of f(x) is: n 1 (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 (Σ1 a f (n)(c) ) Research Volume n=0 n n! 1 Z 1 (x − c)n = p [Σ1 a f (n)(c) ]e−ix!dx Frontier n=0 n 2Π −∞ n! Z 1 n Science 1 1 (n) (x − c) −ix! = p Σn=0anf (c) e dx of 2Π −∞ n! 1 Z 1 1 = p Σ1 a f (n)(c) e−ix!(x − c)ndx Journal n=0 n 2Π −∞ n! 1 1 Z 1 Global 1 (n) −ix! n = p Σ0 anf (c) e (x − c) dx 2Π n! −∞ Let,x = t + c () dx = dt So, n 1 (n) (x − c) F (f(x)) = F (Σn=0anf (c) n! ©2018 Global Journals Fourier Transform of Power Series Z 1 1 1 (n) 1 −i(t+c)! n = p Σ0 anf (c) e t dt 2Π n! −∞ Z 1 1 1 (n) 1 −it! −ic! n = p Σ0 anf (c ) e e t dt 2Π n! −∞ Z 1 1 1 (n) 1 −ic! −it! n = p Σ0 anf (c) e e t dt 2Π n! −∞ otes v 1 N Let, v = it! () t = i! ) dt = i! dv Hence, 1 1 Z 1 v 1 2018 1 (n) −ic! −v n r F (f(x)) = p Σ anf (c) e e [ ] dv 0 ea −∞ 2Π n! i! i! Y Z 1 n 531 1 1 (n) 1 −ic! −v v 1 = p Σ0 anf (c) e e dv 2Π n! −∞ (i!)n i! Z 1 1 1 (n) 1 1 −ic! −v n = p Σ0 anf (c) e e v dv V 2Π n! [i!]n+1 −∞ VII Z 1 1 1 1 ue ersion I 1 (n) −ic! −v n s = p Σ0 anf (c) e [2 e v dv] s n! [i!]n+1 0 2Π I 1 1 (n) 1 1 −ic! Γ(n + 1) XVIII = p Σ anf (c) e [2 ] 2Π 0 n! [i!]n+1 sn+1 2 1 1 Γ(n + 1) 1 (n) = p Σ0 anf (c) ) ic! n! [i!]n+1 sn+1 F 2Πe ( In particular, for n = 1; 2; 3; ::: Research Volume Γ(n + 1) = n! Such that, Frontier n 1 (n) (x − c) 2 1 (n) 1 1 n! F (Σ anf (c) ) = p Σ anf (c) n=0 n! ic! 0 n! [i!]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 1 n f(x) = Σn=0anx Global which is known as Maclaurin series, then the Fourier transform of f(x) is defined by: 1 n F (f(x)) = F (Σn=0anx ) 2 (n) 1 1 Γ(n + 1) = p f (c)Σ an 2Π n=0 (i!)n+1 sn+1 ©2018 Global Journals Fourier Transform of Power Series proof suppose f(x) has the power series expansion at 0 i.e 1 n f(x) = Σn=0anx By using the definition of Fourier transforms, F (f(x)) = F (Σ1 a xn) n=0 n Notes Z 1 1 1 n −ix! = p [Σ anx ]e dx 2018 n=0 2Π −∞ r ea Z 1 Y 1 1 n −ix! = p Σn=0anx e dx 2Π −∞ 541 Z 1 1 1 −ix! n = p Σn=0 ane x dx 2Π −∞ V Z 1 1 1 −ix! n VII = p Σ0 an e x dx 2Π −∞ ue ersion I s s t 1 I Let, t = ix! () x = i! ) dx = i! dt Hence, XVIII Z 1 1 1 −t t n 1 F (f(x)) = p Σ0 an e [ ] dt 2Π −∞ i! i! ) 1 Z 1 tn 1 F 1 −t ( = p Σ0 an e dt 2Π −∞ (i!)n i! Z 1 1 1 1 −t n = p Σ0 an e t dt Research Volume 2Π [i!]n+1 −∞ 1 1 Z 1 p 1 −t n = Σ0 an n+1 [2 e t dt] Frontier 2Π [i!] 0 1 1 1 Γ(n + 1) = p Σ0 an [2 ] Science 2Π [i!]n+1 sn+1 of 2 1 1 Γ(n + 1) = p Σ an 2Π 0 [i!]n+1 sn+1 Journal Note: In particular, for n = 1; 2; 3; Global Γ(n + 1) = n! Hence, 1 n 2 1 1 n! F (Σ anx ) = p Σ an n=0 2Π 0 [i!]n+1 sn+1 ©2018 Global Journals Fourier Transform of Power Series III.