Evaluation of some non-elementary integrals of sine, cosine and exponential integrals type

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1 Noname manuscript No. will be inserted by the editor Evaluation of some non-elementary integrals of sine, cosine and exponential integrals type Victor Nijimbere Received: date / Accepted: date Abstract The non-elementary integrals Si β,α [sin β / α ] and Ci β,α [cos β / α ], where β and α, are evaluated in terms of the hypergeometric functions F and F 3 respectively, and their asymptotic expressions for x are derived. Integrals of the form [sin n β / α ] and [cos n β / α ], where n is a positive integer, are expressed in terms Si β,α and Ci β,α, and then evaluated. On the other hand, Si β,α and Ci β,α are evaluated in terms of the hypergeometric function F. And so, the hypergeometric functions, F and F 3, are expressed in terms of F. The integral Ei β,α e β /x α where β and α, and the logarithmic integral Li x dt/ ln t, are expressed in terms of F, and their asymptotic expressions are investigated. It is found that Li x/ln x+ln ln x ln F, ;, ; ln, where the term ln ln x ln F, ;, ; ln is negligible if x O 6 or higher. Keywords Non-elementary integrals Sine integral Cosine integral Exponential integral Logarithmic integral Hypergeometric functions Asymptotic evaluation Fundamental theorem of calculus Mathematics Subject Classification 6A36 33C5 3E5 Introduction Definition An elementary function is a function of one variable constructed using that variable and constants, and by performing a finite number of re- Victor Nijimbbere School of mathematics and Statistics Carleton University, Ottawa ON Tel.: ext. 37 Fax: nijimberevictor@cmail.carleton.ca

2 Victor Nijimbere peated algebraic operations involving exponentials and logarithms. An indefinite integral which can be expressed in terms of elementary functions is an elementary integral. If it cannot be evaluated in terms of elementary functions, then it is non-elementary[, 6]. Liouville 938 s Theorem gives conditions to determine whether or not a given integral is elementary or non-elementary [,6]. It was shown in [,6], using Liouville 938 s Theorem, that Si, sin x/x is non-elementary. With similar arguments as in [,6], One can show that Ci, cos x/x is also non-elementary. Using the Euler identity e ±ix cos x ± i sin x, and noticing that if the integral of gx is elementary, then both its real and imaginary parts are elementary [], one can readily prove that the integrals Si β,α [sin β / α ] and β, and Ci β,α [cos β / α ], where β and α, are non-elementary by using the fact that their real and imaginary parts are non-elementary. The integrals [sin n β / α ] and [cos n β / α ], where n is a positive integer, are also non-elementary since they can be expressed in terms of Si β,α and Ci β,α. To my knowledge, no one has evaluated these integrals before. To this end, in this paper, formulas for these non-elementary integrals are expressed in terms of the hypergeometric functions F and F 3, which are entire functions on the whole complex space C, and whose properties are well known [3,5]. And therefore, their corresponding definite integrals can be evaluated using the Fundamental Theorem of Calculus FTC. For example, the sine integral Si β,α B A sin β α is evaluated for any A and B using the FTC. On the other hand, the integrals Ei β,α e β /x α and / ln x, can be expressed in terms of the hypergeometric function F. This is quite important since one may re-investigate the asymptotic behavior of the exponential Ei and logarithmic Li integrals using the asymptotic expression of the hypergeometric function F. Other non-elementary integrals can be written in terms of Ei β,α or / ln x. For instance, as a result of substitution, the integral e eβx can be written in terms of Ei β, e β /x and then evaluated, and using integration by parts, the integral lnln x can be written in terms of / ln x and then evaluated. Using the Euler identity e ±ix cosx ± i sinx or the hyperbolic identity e ±x coshx ± sinhx, Si β,α and Ci β,α can be evaluated in terms Ei β,α. In that case, one can express the hypergeometric functions F and F 3 in terms of the hypergeometric F.

3 Title Suppressed Due to Excessive Length 3 Evaluation of the sine integral and related integrals Proposition The function Gx x F ; 3, 3 ; x, where F is a hypergeometric function [] and is an arbitrarily constant, is the antiderivative of the function gx sin. Thus, sin x F ; 3, 3 ; x + C. Proof To prove Proposition, we expand gx as Taylor series and integrate the series term by term. We also use the gamma duplication formula [] Γ α π α Γ αγ α +, α C, 3 the Pochhammer s notation for the gamma function [], α n αα + α + n Γ α + n, α C, Γ α and the property of the gamma function Γ α + αγ α. For example, Γ n + 3 n + Γ n + for any real n. We then obtain sin gx n n+ n +! x n n x n n +! n + + C x Γ n + Γ n + Γ n + 3 n + C n x x n C n! n n x F ; 3, 3 ; x + C Gx + C. 5 Lemma Assume that Gx is the antiderivative of gx sin x x.. Then gx is linear at x and the point, G, is an inflection point of the curve Y Gx.

4 Victor Nijimbere. And lim x constant. Gx θ while lim Proof. The series gx sin x x Gx θ, where θ is a positive finite x + n n n+! gives G g. Then, around x, Gx x since G g. And so,, G,. Moreover G g. Hence, by the second derivative test, the point, G, is an inflection point of the curve Y Gx.. By Squeeze theorem, lim gx lim gx, and since both gx x x + and Gx are analytic on R, Gx has to constant as x ± by Liouville Theorem section 3..3 in [3]. Also, there exists some numbers δ > and ɛ such that if x > δ then sin x /x /x < ɛ, and lim sin x /x//x x lim sin x /x//x ±. This makes the function g x /x an x + envelop of gx away from x if sin x < and g x /x an envelop of fx away from x if sin x >. Moreover, on one hand, g G g if x < δ, and and g and g do not change signs. While on another hand, g G g if x > δ, and also g and g do not change signs. Therefore there exists some number θ > such Gx oscillates about θ if x > δ and Gx oscillates about θ if x < δ. And Gx θ if x δ. Example For instance, if, then sin x x x F ; 3, 3 ; x + C. 6 According to, the antiderivative of gx sin x x is Gx x F ; 3, 3 ; x, and the graph of Gx is in Figure. It is in agreement with Lemma. It is seen in Figure that, G, is an inflection point and that G attains some constants as x ± as predicted by Lemma. Lemma Consider Gx in Proposition, and preferably assume that >.. Then, and G G+. And by the FTC, lim Gx lim x F x x ; 3 ; x π, 7 lim Gx lim x F x + x ; 3 ; x π. 8 sin G+ G π π π. 9

5 Title Suppressed Due to Excessive Length 5.5 θ π.5 Gx.5 θ π Fig. The antiderivative of the function gx sin x x given by 6. x Proof. To prove 7 and 8, we use the asymptotic formula for the hypergeometric function F which is valid for z. It can be derived using formulas 6.., 6.. and 6..8 in [5] and is given by F a ; b, b ; z Γ b Γ b z a { R } a n z n + O z R Γ b a nγ b a n n! Γ b Γ b + ez e i π ze iπ a b b + { S } µ n Γ a π n+ ze iπ n + O z S + Γ b Γ b Γ a e z e i π ze iπ a b b + π { S µ n n+ zeiπ n + O z S where a, b and b are constants and the coefficient µ n is given by formula 6.. in [5]. We then set z x, a, b 3 and b 3, and obtain F ; 3, 3 ; x π x π e i x π e i x { R { S { S n i n + O n! µ n n i n + O µ n n i n + O }, R } S } S},

6 6 Victor Nijimbere Then, for x, while π x π e i x { R { S µ n n { S n n! i n + O i µ n n i n + O n + O π e i + e i cos π. We then obtain, x F ; 3, 3 ; x π x π x Hence, and. By the FTC, + sin } R S } S } π x, cos, x. 3 G lim x F x ; 3, 3 ; x π x π lim x x cos π G+ lim x F x + ; 3, 3 ; x π x π lim x + x cos π. 5 lim y y lim y + y F sin + lim y y + ; 3, 3 ; y G+ G π π sin lim y F y π. 6 ; 3, 3 ; y We now verify wether Lemma this is correct or not. We first observe that + sin + sin, 7

7 Title Suppressed Due to Excessive Length 7 since the integrand is an even function. And write 7 as a double integral. + sin Then by Fubini s theorem [], e sx sin ds + + e sx sin ds. 8 + e sx sin ds. 9 Now using the fact that the inside integral in 9 is the Laplace transform of sin [] yields + + Therefore, e sx sin ds sin + e sx sin ds s + ds arctan + arctan π. + + e sx sin ds π. Hence, + sin + sin π π. as before. This completes the proof. If as in Lemma, Lemma gives lim Gx θ π/ while x Gx θ π/. And these are the exact values of Gx as x ± in lim x + Figure. Theorem If β and α, then the function Gx xβ α+ β α + F α β + β + ; α β + β + 3, 3 ; x β,

8 8 Victor Nijimbere where F is a hypergeometric function [] and is an arbitrarily constant, is the antiderivative of the function gx sin β. Thus, α sin β Si β,α α xβ α+ β α + F α β + β + ; α β + β + 3, 3 ; x β + C. 3 And if β α and α, then sin α α x F α ; α +, 3 ; x α + C. If β and α, then for x, x β α+ β α + F α β + β + ; α β + β + 3, 3 ; x β + β α β Γ α β + β + 3 π x β α+ β α + Γ + α β x β α+ β α + π cos β β x β+α. β 5 And if β α and α, then for x, x F α ; α +, 3 ; x α α Γ α + π x π x Γ 3 α α cos α x α. 6

9 Title Suppressed Due to Excessive Length 9 Proof sin β Si β,α gx α α n β n+ n +! n n x βn+β α n +! n n x βn+β α+ n +! βn + β α + + C xβ α+ n n x βn β n +! n α β + β + + C xβ α+ Γ n α β + β + β n + C β Γ n + Γ n α β + β + 3 n α xβ α+ β + β + x β n β α + + C 3 α n β + β + 3 n! n xβ α+ β α + F α β + β + ; α β + β + 3, 3 ; x β + C Gx + C. 7 Equation is proved by setting β α in 3. To prove 5, we use the asymptotic formula for the hypergeometric function F, given by, and proceed as in Lemma. And 6 is proved by setting β α in 5. One can show as above that if β and α, then sinh β α xβ α+ β α + F α β + β + ; α β + β + 3, 3 ; x β + C. 8 Corollary Let β α. If α, then sin α α G G α Γ α + Γ 3 α π, 9 + sin α α G+ G α Γ α + π Γ 3 α 3

10 Victor Nijimbere and + sin α α G+ G α Γ α + π. 3 Γ 3 α Proof If β α, Theorem gives G lim x F x α ; α +, 3 ; x α α π Γ α lim + x Γ 3 x π x cos α α α x α α Γ α + π Γ 3 α 3 and G+ lim x F x + α ; α +, 3 ; x α α π Γ α lim + x + Γ 3 x x α α Γ α + Γ 3 α π cos α α x α π. 33 Hence, by the FTC, sin α α G G α Γ α + Γ 3 α π α Γ α + Γ 3 α π, 3 + sin α α G+ G α Γ α + Γ 3 α π α Γ α + π Γ 3 α 35

11 Title Suppressed Due to Excessive Length and + sin α α G+ G α Γ α Γ α + Γ 3 α π α Γ α + π Γ 3 α α + π. 36 Γ 3 α Theorem If β and α, then the FTC gives B A sin β α GB GA, 37 for any A and any B, and where G is given in 3. Proof Equation 37 holds by Theorem, Corollary and Lemma. Since the FTC works for A and B in 9, A and B + in 3 and A and B + in 3 by Corollary for any β α and by Lemma for β α, then it has to work for other values of A, B R and for β and α since the case with β α is derived from the case with β and α. 3 Evaluation of the cosine integral and related integrals Theorem 3 If β, then the function Gx β ln x β F 3, ; 3,, ; x β, where F 3 is a hypergeometric function [] and is an arbitrarily constant, is the antiderivative of the function gx cos β. Thus, cos β ln x β β F 3, ; 3,, ; x β β + C. 38

12 Victor Nijimbere Proof cos β gx n β n n! + n n n! xβn n ln x n n+ x βn+β n +! ln x n n x βn+β n +! βn + β + C β Γ n + ln x β Γ n + 3Γ n + x β n + C n β ln x n x β n β 3 n + C n n n! β ln x β F 3, ; 3,, ; x β + C Gx + C. 39 Theorem If β and α >, then the function x α Gx α β α+ β α + F 3, α β + β + ; α β + β +, 3, ; x β where F 3 is a hypergeometric function [] and is an arbitrarily constant, is the antiderivative of the function gx cos β. Thus, α cos β α x α α β α+ β α + F 3, α β + β + ; α β + β +, 3, ; x β + C. And for x, β α+ β α + F 3 π β Γ, α β + β + ; α β + α β + β + α β +, 3, ; x β β + β + + π β x α+ + β cos β x β+α.,

13 Title Suppressed Due to Excessive Length 3 Proof If β and α >, cos β gx α α n β n n! α + n n n! xβn α n x α α n n+ x βn+β α n +! x α α n n x βn+β α+ n +! βn + β α + + C x α α β α+ Γ n α β + β + x β n + C β Γ n + 3Γ n α β + β + n x α α β α+ β α + n α β + β n + x β + C 3 n n α β + β n + n! x α α β α+ β α + F 3, α β + β + ; α β + β +, 3, ; x β + C Gx + C. To prove, we use the asymptotic expression of the hypergeometric function F 3 a, a ; b, b, b 3 ; z for z, where a, a, b, b and b 3 are constants. It can be obtained using formulas 6.., 6.. and 6..8 in [5] and is

14 Victor Nijimbere given by F 3 a, a ; b, b, b 3 ; z Γ b Γ b Γ b 3 z a Γ a { R } Γ a a n z n a n + O z R Γ b a nγ b a nγ b 3 a n n! { R a n Γ a a n Γ b a nγ b a nγ b 3 a n ez e i π ze iπ a +a b b b 3 + π ez e i π ze iπ a +a b b b 3 + π + Γ b Γ b Γ b 3 z a Γ a } z n + O z R n! + Γ b Γ b Γ b 3 Γ a Γ a { S } µ n n+ ze iπ n + O z S { S µ n + + Γ b Γ b Γ b 3 Γ a Γ a } n+ zeiπ n + O z S 3, where the coefficient µ n is given by formula 6.. in [5]. Setting z x β, a, a α β + β +, b α β + β +, b 3 and b 3 in 3 yields F 3, α β + β + ; α β + β +, 3, ; x β π α β + π β + x β + Γ α β + α β + β + β + β + 3 α β + cos β β + x 3β. This gives β α+ β α + F 3 π β α + Γ π β Γ, α β + α β + β + α β + β + β + ; α β + β +, 3, ; x β α β + β + π + β x α+ + β α which is exactly. This completes the proof. β + β + + π β x α+ + β cos β x β+α cos β x β+α,

15 Title Suppressed Due to Excessive Length 5 One can show as above that if β and α >, then cosh β α x α α β α+ + β α + F 3, α β + β + ; α β + β +, 3, ; x β + C. 5 Evaluation of some integrals involving Si α,β and Ci α,β The integral cos n β, where n is a positive integer and β, α, can α be written in terms of and then evaluated. Example In this example, the integral cos β is evaluated by linearizing the function cos β. This α gives cos β α cos β 8 cos β α α + x α 8 α β α+ β α + F 3, α β + β + ; α β + β +, 3, ; x β + x α α β α+ β α + F 3, α β + β + ; α β + β +, 3, ; x β + 3x 8 + C 6 If β and α, the integral sin n β, where n is a positive integer, α can be written either in terms of 3 if n odd, or in terms of if n even, and then evaluated. Example 3 In this example, the integral sin 3 β is evaluated by linearizing the function sin 3 β. This α gives sin 3 β α sin 3 β x β α+ β α + F α + 3 sin β α α β + x β α+ + 3 β α + F β + ; α β + β + 3, 3 ; x β 9 α β + β + ; α β + β + 3, 3 ; x β 7 + C

16 6 Victor Nijimbere Example In this example, the integral sin β is evaluated by linearizing the function sin β. This α gives sin β α 8 x α 8 α x α β α+ cos β α, α β + β α + F 3 β α+ α β α + F 3 cos β α 3 8 β + ; α β + β +, 3, ; x β, α β + β + ; α β + β +, 3, ; x β 3x 8 + C 8 The integrals sin x and cos µ x, where the constant µ can µ be evaluated by making the substitution u /x. Example 5. Making the substitution u /x and applying Theorem gives sin u µ sin x µ u uµ µ F µ + ; µ + 3, 3 ; u µ µ x F µ µ + ; µ + 3, 3 ; x µ + C 9. Making the substitution u /x and applying Theorem gives cos u µ cos x µ u u + uµ µ F 3, µ + ; µ +, 3, ; u µ x + µ x F 3, µ µ + ; µ +, 3, ; x µ + C 5 5 Evaluation of exponential Ei and logarithmic Li integrals Proposition If β, then for any constant, and e β β ln x + x β F, ;, ; β + C, 5 F, ;, ; + eβ, x. 5 β

17 Title Suppressed Due to Excessive Length 7 Proof e β x β n x n! x + n x βn n! n n ln x + x βn n! n n x βn ln x + n! βn n n+ x βn+β ln x + n +! βn + β ln x + β Γ n + β Γ n + Γ n + β n + C ln x + β β n n β n + C n n n! ln x + β β F, ;, ; β + C. 53 To derive the asymptotic expression of β F, ;, ; β, x, we use the asymptotic expression of the hypergeometric function F a, a ; b, b ; z for z, where z C, and a, a, b and b are constants. It can be obtained using formulas 6.., 6.. and 6..7 in [5] and is given by F a, a ; b, b ; z Γ b Γ b Γ a { R ze ±iπ a a n Γ a a n ze ±iπ n + O z R Γ b a nγ b a n n n! ze ±iπ a { R + Γ b Γ b Γ a a n Γ a a n ze ±iπ n Γ b a nγ b a n n n! + Γ b Γ b Γ a Γ a ez z a +a b b { S + O z R µ n n z n + O z S } } }, 5 where the coefficient µ n is given by formula 6... And the upper or lower signs are chosen according as z lies in the upper above the real axis or lower half-plane below the real axis.

18 8 Victor Nijimbere Hence, Setting z β, a, a, b and b in 5 yields F, ;, ; β β + e β, x. 55 xβ β F, ;, ; + eβ, x. 56 β This ends the proof. Example 6 One can now evaluate e eβx in terms of F using the substitution u e x, and obtain e u β e eβx uβ du ln u + u β F, ;, ; u β + C x + eβx β F, ;, ; e βx + C. 57 Theorem 5 Li x dt ln t ln And for x, Li x ln x dt ln t x ln x + ln + ln x F, ;, ; ln x ln F, ;, ; ln. ln x 58 ln F, ;, ; ln. 59 Proof Making the substitution u ln x and using 5 gives x ln x ln x ln e u u du [ln u + u F, ;, ; u] ln x ln ln x ln + ln x F, ;, ; ln x ln F, ;, ; ln. 6 Now setting z ln x, a, a, b and b in 5 yields This gives F, ;, ; ln x ln x + x, x. 6 ln x ln x F, ;, ; ln x + x, x. 6 ln x

19 Title Suppressed Due to Excessive Length 9 Hence for x, Li x dt ln t x ln x + ln ln x ln F, ;, ; ln. 63 It is important to note that Theorem 5 adds the term ln ln x ln F, ;, ; ln to the known asymptotic expression of the logarithmic integral 65, Li x/ln x [,5]. And this term is negligible if x O 6 or higher. Example 7 One can now evaluate ln ln x using integration by parts. ln ln x x ln ln x ln x x ln ln x ln ln x ln x F, ;, ; ln x + C. 6 Theorem 6 If β and α >, then e β x α Ei α,β x α α + β α+ β α + F And for x,, αβ + β + ;, αβ + β + ; β +C. 65 β α+ β α + F, αβ + β + ;, αβ + β + ; β Γ α β + β + α β + β + x α+ β + β e β. 66 xβ+α

20 Victor Nijimbere Proof If β and α >, then e β Ei β,α x α x α α + n x α α + n n! x α n+ x α α + β α+ β x α α + β α+ β α + β n n! x βn α x α α + β α+ β α + F x βn+β α+ n +! βn + β α + + C Γ n αβ + β + Γ n + Γ n β n + C αβ + β + n αβ + β + n β n + C n α β + β + n! n, αβ + β + ;, αβ + β + ; β + C. 67 Now setting a, a α β + β +, b, b α β + β + and z β in 5 gives, F, α β + β + ;, α β + β + ; β α β + β + β + Γ α β + α β + β + β + β + eβ x β. 68 Hence, β α+ β α + F, αβ + β + ;, αβ + β + ; β β α + Γ αβ + β + α β + β + x α+ β β Γ α β + β + which is 66. This completes the proof. α β + β + x α+ β + β + β e β x β+α e β x β+α,

21 Title Suppressed Due to Excessive Length Theorem 7 For any constants α, β and, F α β + β + ; α β + β + 3, 3 ; x β [F, α β + β + ;, α β + β + ; iβ + F, α β + β + ;, α β + β + ; iβ ]. 69 Or, F α β + β + ; α β + β + 3, 3 ; x β [F, α β + β + ;, α β + β + ; β + F, α β + β + ;, α β + β + ; β ]. 7 Proof Using Theorem 6, we obtain sin β x α e i β e iβ i x α β α+ [ F, α β α + β + β + ;, α β + β + ; iβ + F, α β + β + ;, α β + β + ; iβ ] + C. 7 Hence, comparing 3 with 7 gives 69. Or, on the other hand, sinh β x α β α+ β α + e β e β x α [ F, α β + β + ;, α β + β + ; β + F, α β + β + ;, α β + β + ; β ] + C. 7 Hence, comparing 8 with 7 gives 7.

22 Victor Nijimbere Theorem 8 For any constants α, β and, x β α+ β α + F 3, α β + β + ; α β + β +, 3, ; x β x β α+ [ F, α i β α + β β + ;, α β + β + ; iβ F, α β + β + ;, α β + β + ; iβ ]. 73 Or, x β α+ β α + F 3, α β + β + ; α β + β +, 3, ; x β xβ α+ β α + [ F, α β β + ;, α β + β + ; β + F, α β + β + ;, α β + β + ; β ]. 7 Proof Using Theorem 6, we obtain cos β x α x α α i β α + e i β + e iβ β α+ x α [ F, α β + β + ;, α β + β + ; iβ F, α β + β + ;, α β + β + ; iβ ] + C, 75 Hence, comparing with 75 gives 73. Or, on the other hand, cosh β x α x α α + β α + e β + e β β α+ x α [ F, α β + β + ;, α β + β + ; β F, α β + β + ;, α β + β + ; β ] + C, 76 Hence, comparing 5 with 76 gives 7.

23 Title Suppressed Due to Excessive Length 3 6 Conclusion Si β,α [sin β / α ] and Ci β,α [cos β / α ], where β and α, were expressed in terms of the hypergeometric functions F and F 3 respectively, and their asymptotic expressions for x were obtained see Theorems, 3 and. Ei β,α e β /x α and / ln x were expressed in terms of the hypergeometric function F, and their asymptotic expressions for x were also obtained see Proposition, Theorems 5 and 6. Therefore, their corresponding definite integrals can now be evaluated using the FTC see for example Corollary. Using the Euler and hyperbolic identities Si β,α and Ci β,α were expressed in terms of Ei β,α. And hence, some expressions of the hypergeometric functions F and F 3 in terms of F were derived see Theorems 7 and 8. The evaluation of the integral / ln x and the asymptotic expression of the hypergeometric function F for x allowed us to add the term ln ln x ln F, ;, ; ln to the known asymptotic expression of the logarithmic integral, which is Li x dt/ ln t x/ln x [,5], so that it is given by Li x dt/ln t x/ln x + ln ln x ln F, ;, ; ln Theorem 5. By substitution, the non-elementary integral e eβx was written in terms of Ei β, and then evaluated in terms of F. And using integration by parts, the non-elementary integral lnln x was written in terms of / ln x and then evaluated in terms of F. References. M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs and mathematical tables, National Bureau of Standards, 96.. P. Billingsley, Probability and measure, Wiley series in Probability and Mathematical Statistics, S. G. Krantz, Handbook of Complex variables, Boston, MA Birkäusser, E. A. Marchisotto, G.-A. Zakeri, An invitation to integration in finite terms, College Mathematical Journal 5: NIST Digital Library of Mathematical Functions, 6. M. Rosenlicht, Integration in finite terms, American Mathematical Monthly 79: