M. A. Pathan, O. A. Daman LAPLACE TRANSFORMS OF THE LOGARITHMIC FUNCTIONS AND THEIR APPLICATIONS

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1 DEMONSTRATIO MATHEMATICA Vol. XLVI No 3 3 M. A. Pthn, O. A. Dmn LAPLACE TRANSFORMS OF THE LOGARITHMIC FUNCTIONS AND THEIR APPLICATIONS Abtrct. Thi pper del with theorem nd formul uing the technique of Lplce nd Steiltje trnform expreed in term of intereting lterntive logrithmic nd relted integrl repreenttion. The dvntge of the propoed technique i illutrted by logrithm of integrl of importnce in certin phyicl nd ttiticl problem.. Introduction The im of thi pper i to obtin ome theorem nd formul for the evlution of finite nd infinite integrl for logrithmic nd relted function uing technique of Lplce trnform. Bic propertie of Lplce nd Steiltje trnform nd Prevl type reltion re explicitly ued in combintion with rule nd theorem of opertionl Clculu. Some of the integrl obtined here re relted to Stochtic clculu [6] nd common mthemticl object, uch the logrithmic potentil [3] logrithmic growth [] nd Whittker function [, 3, 4, 6, 7] which re of importnce in certin phyicl nd ttiticl ppliction, in prticulr in energie, entropie [3, 5, 7 )], intermedite moment problem [] nd quntum electrodynmic.the dvntge of the propoed technique i illutrted by the explicit computtion of number of different type of logrithmic integrl. We recll here the definition of the Lplce trnform ) L {f t)} = L[ft); ] = e t f t) dt. Cloely relted to the Lplce trnform i the generlized Stieltje trnform Mthemtic Subject Clifiction: 33C5, 33C9, 44A. Key word nd phre: Lplce nd Steiltje trnform, logrithmic function nd integrl, tochtic clculu nd Whittker function.

2 538 M. A. Pthn, O. A. Dmn ) S ρ {ft)} = ft)dt = G, ρ) + t) ρ which for ρ = give the Stieltje trnform ft)dt 3) S{ft)} = + t) = G). After chnge of integrtion vrible, ) i 4) f ln x) x ln x) ρ dx = ρ G, ρ). Specil ce of ) nd 4), when ft) = t λ e t, re generliztion of gmm function given by Kobyhi [7] 5) Γ ρ λ, ) = ln x) λ ln x) ρ dx, Rλ) >. Kobyhi [7] pplied thi generlized gmm function integrl in diffrction theory.. Theorem In thi Section, we tte nd prove ome theorem in the tudy of integrl trnform, nd briefly dicu ome pprent, known nd new pecil ce of thee theorem. We will pply ytemticlly the rule nd theorem of the opertionl clculu uming the exitence of the Lplce trnform of the function involved nd the permiibility of performed mthemticl opertion. Theorem. If 6) L{ft)} = ϕ) nd 7) L{ht)} = g) then 8) 9) ) ln x n ) ln x g f = ϕt)h n t )dt = ln x ϕ + x)h n x)dx ) dx

3 ) nd ) Lplce trnform of the logrithmic function nd their ppliction 539 L[{h n t )Ht )}; ]d = Γn + ) φ ln x u ) u ln x) dx = u L[φt )Ht ); x]dx h n t) dt t + ) n+ provided tht ft) L, ), e t t n gt) L, ), ht) L, ) nd h n t) denote the n th differentil coefficient of ht) uch tht h ) = h ) =... = h n ) =. Ht) i Heviide unit function nd integrl in 8) to ) re convergent. Proof. Conider the Lplce trnform ) L{ft)} = L[ft); ] = e t ft)dt = ϕ). Applying well known property of the Lplce trnform [4, p. 9] tht i, if 3) L{ht)} = g) then 4) L{h n t)} = n g), h ) = h ) = = h n ) = nd 5) L{h n t )Ht )} = e n g), where Ht) i Heviide unit function. To prove 8) nd 9), we ue ) nd 5) in the Prevl theorem nd then by chnging the integrtion vrible x = e t, we find ln x ) n g ln x ) f ln x ) 6) dx = ϕt)h n t )dt = ϕ + x)h n x)dx, where i replced by. Now we integrte both ide of 5) nd ue 4) to obtin 7) L{h n t )Ht )}d [ ] = e n e t h n t)dtd = h n t) n e t d dt.

4 54 M. A. Pthn, O. A. Dmn By evluting the integrl on the right hnd ide of 7) we obtin Prevl reltion for 3). Thu 8) L{h n h n t) t )}Ht )}dt = Γn + ) dt. t + ) n+ For n =, 8) give L{ht )Ht )}dt = =, 8) give known reult [5, p..4)] ht) 9) L{ht)}d = dt. t ht) t+) dt. For n = nd Now, et ht) = e ut φt) nd n = in 9) nd ue hift property ) L{ht)} = L{e ut φt)} = e ut t φt)dt = L[φt); u + ] to get nother Prevl-type reltion [ ] φt) ) L[φt )Ht ); x]dx = L + t ; u, u by mking the chnge of vrible u + = x. On chnging the integrtion vrible x = e ut, we get ). For =, ) give known reult [5, p..8)] [ ] φt) ) L[φt); x]dx = L ; u. t If we tke ht) = t λ e t nd ue [4, p. 9] u 3) L{t n ft)} = ) n dn F ), n =,,... dn nd binomil theorem in Theorem, we obtin Theorem. If L[ft); ] = F ) then ) 4) x ln x) n ln x n+ dn f dx = F + ) dn nd 5) ln x ) n ln x ) λ f ln x ) dx ) n ) n r n r n = φ; ) Γλ r + ) r r=

5 Lplce trnform of the logrithmic function nd their ppliction 54 where 6) φ; ) = x λ r e x F + x)dx provided tht Lplce trnform of ft) exit, λ > n, R + ) > nd the integrl in 5) nd 6) re convergent. In the implet ce ft) = nd F ) =, we hve immeditely from 5) nd [4, p. 94 6)] 7) ln x ) n ln x ) λ dx = e ) n ) n r n λ+r+)/ n λ r )/ W k,m ) r r= where k = r λ )/, m = λ r)/ nd W k,m x) i Whittker function [4]. Evidently, if we et ft) = in 4), we get ) + n 8) x / ln x) n dx = ) n n!. Another exmple i [ ] L[ft)] = L + e t = [ ψ which led to 9) x / ln x) n + x / = n+ n+ + ) ψ )], R) > [ ) )] ψ n) ψ n) n =,,..., R) >, where ψζ) i pi-function nd ψ n) ζ) men the n th derivtive of pi-function [4]. On the other hnd, the pecil ce of 4) nd 5), for n = nd = yield known reult [, p. 4,eqution 3), 6) nd 8)]. Theorem 3. Let α >, β >, then 3) f ln x α+β ) ln/x) dx = παα + β)l[gθ, t); α] where t e βθ fθ) 3) gθ, t) = π θt θ) dθ

6 54 M. A. Pthn, O. A. Dmn Proof. Let gθ, t) be defined by 3). Then [ t L[gθ, t); α] = e αt e βθ ] [ fθ) π θt θ) dθ e βθ dt = π θ fθ) θ [ ] = e α+β)θ e α fθ) d dθ π θ ] e αt dt dθ t θ where in the inner integrl, we hve chnged the vrible of integrtion by etting t θ =. It follow from e τt π 3) dt = t τ, Rτ) > then L[gθ, t); α] = πα e α+β)θ θ fθ)dθ. The uniquene of Lplce trnform nd the ubtitution e α+β)θ = x implie the required reult. It will be hown tht if we et fθ) = in the integrl 3) nd ue 3), then theorem 3 reduce to the following P. Levy Arc-Sine Lw for occuption time of, ) [6, p. 73, Art 4.]. Let α >, β >. Then 33) L[hθ, t); α] = αα + β) where 34) hθ, t) = π t e βθ θt θ) dθ. Auming the exitence of the Lplce trnform of ft), we conider L{ft)} = F ) nd then uing the rule of Lplce trnform L{ft + )} = e [F ) e u fu)du], nd 35) L{e bt ft + )} = e +b) [F + b) e +b)u fu)du], we get the following theorem. Theorem 4. If L{ft)} = F ), then 36) e b ln x)/ f ln x [ )dx = e +b) F + b) ] e +b)u fu)du

7 Lplce trnform of the logrithmic function nd their ppliction 543 provided tht, R) >, Lplce trnform of ft) exit nd integrl in 36) re convergent. Eqution 36) i generliztion of the reult [, p. follow for b =. Theorem 5. If then 37) S ρ {gt)} = G; ρ) x g ln x) ln x) ρ dx = e ρ G; ρ). 39 )] which Proof. The bove theorem cn be proved eily if the definition integrl ) i pplied in the form S ρ {e t e t gt) gt)} = + t) ρ dt which fter chnge of integrtion vrible nd mere integrtion by prt together with 36) led to x g ln x) ln x) ρ dx = ρ [e )G; ρ) + e t G; ρ)dt] = ρ [e )G; ρ) + G; ρ)] Prticulrly, for = 37) reduce to [, p. 5, eqution 8b)]. 3. Appliction A number of ppliction of the formul for the evlution of finite nd infinite logrithmic integrl uing the opertionl clculu technique of Section cn be given. We lit ome of them. A n exmple of 36), we tke ft) = / t o tht L{/ t} = π = F ) nd 36) give [ ] b ln x)/ dx π 38) e = e +b) ln x/ + b erf + b)) where erf ξ i the error function [4]. For b =, 38) become known reult[, p. 4 4)]. Put α = nd β = b/ in 36) nd ue erf α = erfc α where α > to get 39) e β ln x dx π = e α+β) α ln x + β erfc α + β) becue erfξ then become the complementry error function erfcξ ee [4]).

8 544 M. A. Pthn, O. A. Dmn In the econd ce we trt with ft) = ln t nd we ue 36) to get 4) e b ln x)/ ln ln x )dx = e+b) [ ] e +b) ln Ei b) + b where τ i the Euler contnt [4] nd Eiξ) i the exponentil integrl [4]. Uing the Stieltje trnform S ρ {e t } = e Γ ρ, ), Rρ) > where Γ. ) i n incomplete gmm function in Theorem 5, we get 4) x + dx ln x) ρ = ρ S ρ {e t+) } For =, we get 4) = [ + )] ρ e +) Γ ρ, + )), >, ρ >. x dx ln x) ρ = ρ e Γ ρ, ), >, ρ > which i correct form of the reult [, p. 5, 97b)]. 4) become known reult [, p. 5, 97)] when ρ =. Next, we will turn our ttention to the ce when gt) = te t in 37). Thu we hve from [8, p )] 43) lnx / )dx ln x) = πe Γ, ). Formul 4) to 4) how tht more generl ce cn be conidered by uing gt) = t λ e t in Theorem 5. By defining generliztion of gmm function ee Kobyhi [7]) 44) Γ ρ, λ, ) = t λ e t t + ) ρ dt, Rλ), R) > nd chnging the integrtion vrible, we hve 45) Γ ρ, λ, ) = λ ln x ) ρ ln x) λ dx Rλ), R) > which for = reduce to the known generliztion of gmm function 5) given by Kobyhi [7]. Note tht Γ ρ, λ, ) = Γ ρ λ, ). In view of the reult [4, p.94 6)], it i more nturl to work with 37) nd 45) to obtin 46) Γ ρ, λ, ) = Γλ)) ρ λ )/ e / W k,m, ) where k = λ ρ, m = λ ρ nd Rλ) >. Thu, we hve provided n integrl repreenttion for Γ ρ, λ, ) in term of Whittker function.

9 Lplce trnform of the logrithmic function nd their ppliction 545 We now ue theorem, reult 4) to obtin generliztion of the reult of Apelblt [,p38,4)] involving the n th derivtive of pi-function x b/ ln x) n ) n+ ) + b 47) dx = ϕ n). x/ For b =, 47) give corrected form of the reult [, p. 38 4)] ln x) n 48) x α dx = ) ϕn) α) n+ α where α = /. To prove 47), notice tht for n =,,..., { } ) t n 49) L{ft)} = L e t = ) n ϕ n) = F ) which i well-known reult ee [4] nd [, p. 38 3)]. From the opertionl reltion 4) nd 49), reult 47) follow. Reference [] A. Apelblt, Appliction of Lplce trnformtion to evlution of integrl, J. Mth. Anl. Appl ), [] C. Berg, H. L. Pederen, Logrithmic order nd type of intermedite moment problem, Proceeding of the Interntionl Conference on Difference Eqution, Specil Function nd Orthogonl Polynomil, Munich, Germny, 5 3 July 5, [3] J. S. Dehe, A. Mtinerz-Finkelhtein, J. Snchez-Ruiz, Quntum informtion entropie nd orthogonl polynomil, J. Comput. Appl. Mth. 33 ), [4] A. Erdelyi et l., Tble of Integrl Trnform, Vol., McGrw Hill, New York, 954. [5] S. Hermn, J. Mceli, S. Rogl, O. Yurekli, Prevl type reltion for Lplce trnform nd their ppliction, Internt. J. Mth. Ed. Sci. Tech. 39 ) 8), 9 5. [6] I. Krtz, S. E. Shreve, Brownin Motion nd Stochtic Clculu, Springer, New York, 99. [7] K. Kobyhi, On generlized gmm function occurring in diffrction theory, J. Phy. Soc. Jpn 6 99), 5 5. [8] J. S. Lomont, J. Brillhrt, Elliptic Polynomil, Chpmn nd Hll, New York, 98. DEPARTMENT OF MATHEMATICS UNIVERSITY OF BOTSWANA P/BAG, GABORONE, BOTSWANA E-mil: mpthn@gmil.com; dmno@mopip.ub.bw Received Augut,.