EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART II

Transcription

1 URAL MATHEMATICAL JOURNAL, Vol., No., 208, pp DOI: /umj EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART II Victor Nijimbere School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada victornijimbere@gmail.com Abstract: The non-elementary integrals Si β,α [sinλx β /λx α ]dx, β, α > β and Ci β,α [cosλx β /λx α ]dx, β, α > 2β, where {β,α} R, are evaluated in terms of the hypergeometric function 2 F 3. On the other hand, the exponential integral Ei β,α /x α dx, β, α > β is expressed in terms of 2 F 2. The method used to evaluate these integrals consists of expanding the integrand as a Taylor series and integrating the series term by term. Key words: Non-elementary integrals, Sine integral, Cosine integral, Exponential integral, Hyperbolic sine integral, Hyperbolic cosine integral, Hypergeometric functions.. Introduction Let us first give the definition of the non-elementary integral. This definition is also given in Part I [6], we repeat it here for reference. Definition. An elementary function is a function of one variable constructed using that variable and constants, and by performing a finite number of repeated algebraic operations involving exponentials and logarithms. An indefinite integral which can be expressed in terms of elementary functions is an elementary integral. And if, on the other hand, it cannot be evaluated in terms of elementary functions, then it is non-elementary [, 9]. The cases consisting of the non-elementary integrals Si β,α [sinλx β /λx α ]dx, β, α β and Ci β,α [cosλx β /λx α ]dx, β, α 2β, where {β,α} R, were considered and evaluated in terms of the hypergeometric functions F 2 and 2 F 3 in Part I [6], and their asymptotic expressions for x were derived too in Part I [6]. The exponential integral Ei β,α /x α dx where β and α β was expressed in terms of 2 F 2, and its asymptotic expression for x was derived as well in Part I [6]. Here, we investigate other cases which were not treated neither in Part I [6] nor elsewhere. We evaluate Si β,α [sinλx β /λx α ]dx, β, α > β and Ci β,α [cosλx β /λx α ]dx, β, α > 2β and Ei β,α /x α dx, β, α > β. In order to take into account all possibilities, we write these integrals as Si β,βα [sinλx β /λx βα ]dx, β, α >, Ci β,2βα [cosλx β /λx 2βα ]dx, β,α >, and Ei β,βα /x βα dx, β, α > where {β,α} R. On one hand, Si β,βα and Ci β,2βα are expressed in terms of the hypergeometric function 2 F 3, while on another hand, Ei β,βα is expressed in terms of the hypergeometric function 2 F 2. These integrals involving a power function x β in the argument of the numerator are the generalizations of the exponential, sine and cosine integrals in [7] see sections 8.9 and 8.2 respectively,

2 Victor Nijimbere which have applications in different fields in science, applied sciences and engineering including physics, nuclear technology, mathematics, probability, statistics, and so on. For instance, the generalized exponential integral E,α is used in fluidodynamics and transport theory, where it is applied to the solution of Milne s integral equations [2], there are also used in modeling radiative transfer processes in the atmosphere and in nuclear reactors [0], etc. Exponential asymptotics involving generalized exponential integrals are used in probability theory, see for example [3]. On the hand, generalized sine and cosine integrals are frequently utilized in Fourier analysis and related domains [8]. Therefore, we are justified to further generalize these functions and their connections to hypergeometric functions. Before we proceed to the main objectives of this paper consisting of evaluating the above interesting cases of non-elementary integrals see sections 2, 3 and, we first define the generalized hypergeometric function as it is an important tool that we are going to use in the paper. Definition 2. The generalized hypergeometric function, denoted as p F q, is a special function given by the series [, 7] pf q a,a 2,,a p ;b,b 2,,b q ;x a n a 2 n a p n x n b n b 2 n b q n n!, where a,a 2,,a p and ;b,b 2,,b q are arbitrary constants, ϑ n Γϑ n/γϑ Pochhammer s notation [, 7] for any complex ϑ, with ϑ 0, and Γ is the standard gamma function []. 2. Evaluation of the sine integral Si β,βα, β, α > Theorem. Let β and α >, and let α mβ ǫ, where m is an integer m N and β < ǫ < β.. If ǫ 0, then Si β,βα sinλx β λx βα m λ 2m x 2 m πγmγm3/22β 2 F 3 where m α/β. 2. If ǫ, then sinλx β λ Si β,βα dx m λxβα λ 2n2m x 2βn nm Γ2n2m2 2βn, 2β ;m,m 3 2,2 x 2β 2β ; λ2 2m Γ2m2 ln x m λ 2m2 x 2β 2 2m πγm2γm5/2β 2 F 3 where m α /β. nm λ 2n2m Γ2n2m2,;m2,m 52 x 2β,2; λ2 C, C, x 2βn 2βn

3 Some non-elementary integrals of sine, cosine and exponential integrals type 5 3. Finally, if ǫ β,0 0,,β, we have sinλx β λ 2m x ǫ m λxβα Γ2m2 ǫ nm λ 2n2m x 2βn ǫ Γ2n2m2 2βn ǫ where m α ǫ/β. m λ 2m2 x 2β ǫ 2 2m3 πγm2γm5/22β ǫ 2 F 3, ǫ 2β ;m2,m 5 ǫ x 2β,2 2 2β ; λ2 C, 2.3 P r o o f. We proceed as in [5, 6]. We expand gx as Taylor series and integrate the series term by term. We use the gamma duplication formula[], the gamma property Γα αγα and Pochhammer s notation see Definition 2. We also set α mβ ǫ, and then we obtain sinλx β λxβα λx β x α nλxβ 2n 2n! n λ 2n 2n! x2βn α dx m n λ 2n 2n! x2βn 2βm ǫ dx n λ 2n 2n! x2βn 2βm ǫ dx m nm n λ 2n 2n! x2βn m ǫ dx n λ 2n 2n! x2βn m ǫ dx nm nm λ 2n2m 2n2m! x2βn ǫ dx nm λ 2n2m 2n2m! x2βn ǫ dx nm λ 2n2m Γ2n2m2 x2βn ǫ dx nm λ 2n2m Γ2n2m2 x2βn ǫ dx m λ 2m dx Γ2m2 x ǫ nm λ 2n2m Γ2n2m2 x2βn ǫ dx nm λ 2n2m Γ2n2m2 x2βn ǫ dx m λ 2m Γ2m2 m λ 2m Γ2m2 n nm m λ 2m Γ2m2 dx x ǫ nm dx x ǫ λ 2n2m nm Γ2n2m2 x2βn ǫ dx λ 2n2m2 Γ2n2m x2βn2β ǫ dx λ 2n2m2 λ 2n2m x 2βn ǫ nm Γ2n2m2 2βn ǫ x 2βn2β ǫ Γ2n2m 2βn2β ǫ C dx x ǫ λ 2n2m x 2βn ǫ nm Γ2n2m2 2βn ǫ 2.

4 6 Victor Nijimbere m λ 2m2 x 2β ǫ n ǫ/2β n λ 2 x 2β / n 2 2m3 C πγm2γm5/22β ǫ m2 n m5/2 n 2 ǫ/2β n n! m λ 2m dx Γ2m2 x ǫ nm λ 2n2m x 2βn ǫ Γ2n2m2 2βn ǫ m λ 2m2 x 2β ǫ 2 2m3 πγm2γm5/22β ǫ 2 F 3, ǫ 2β ;m2,m 5 ǫ x 2β,2 2 2β ; λ2 C.. For ǫ 0, we substitute ǫ 0 in 2., and hence, we obtain sinλx β λxβα λx β x α nλxβ 2n 2n! dx nm λ 2n2m Γ2n2m2 x2βn dx nm λ 2n2m Γ2n2m2 x2βn dx nm λ 2n2m x 2βn Γ2n2m2 2βn nm λ 2n2m x 2βn Γ2n2m2 2βn nm Γ2n2m2 2βn m λ 2m x n /2β n λ 2 x 2β / n 2 m π2β ΓmΓm3/2 m n m3/2 n 2/2β n n! m λ 2m x 2 m πγmγm3/22β 2 F 3 which is 2., and where m α/β. λ 2n2m λ 2n2m x 2βn x 2βn nm Γ2n2m2 2βn, 2β ;m,m 3 2,2 x 2β 2β ; λ2 2. For ǫ, we set ǫ in 2 and obtain sinλx β λ 2m m λxβα Γ2m2 ln x nm λ 2n2m Γ2n2m2 m λ 2m2 x 2β 2 2m πγm2γm5/2β 2 F 3,;m2,m 52 x 2β,2; λ2 which is 2.2, and where m α /β. x 2βn 2βn C, C, 3. For ǫ β,0 0,,β, 2 gives sinλx β λ 2m x ǫ m λxβα Γ2m2 ǫ nm λ 2n2m x 2βn ǫ Γ2n2m2 2βn ǫ m λ 2m2 x 2β ǫ 2 2m3 πγm2γm5/22β ǫ 2 F 3, ǫ 2β ;m2,m 5 ǫ x 2β,2 2 2β ; λ2 C,

5 Some non-elementary integrals of sine, cosine and exponential integrals type 7 which is 2.3, and where m α ǫ/β. Example. In this example, we evaluate [ sinx 2 /x 3.5] dx. We first observe that λ and β 2. We also have 3.5 β α β mβ ǫ, and so m and ǫ 0.5. Substituting λ,β 2,m and ǫ 0.5 in 2.3 gives sinx 2 x 3.5 x.5 9 x x5.5 50π 2 F 3, 9 8 ;3, 7 2, 7 8 ; x C. We can use the same procedure for the hyperbolic sine integral, the results are stated in the following theorem. Its proof is similar to that of Theorem, we will omit it. Theorem 2. Let β and α >, and let α mβ ǫ, where m is an integer m N and β < ǫ < β.. If ǫ 0, then sinhλx β λx βα λ 2m x 2 m πγmγm3/22β 2 F 3 where m α/β. 2. If ǫ, then λ 2n2m sinhλx β λ 2m λxβα Γ2m2 ln x λ 2m2 x 2β 2 2m πγm2γm5/2β 2 F 3 where m α /β. 3. Finally, if ǫ β,0 0,,β, we have sinhλx β λ 2m x ǫ λxβα Γ2m2 ǫ where m α ǫ/β. x 2βn Γ2n2m2 2βn, 2β ;m,m 3 2,2 2β ; λ2 x 2β λ 2n2m x 2βn Γ2n2m2 2βn,;m2,m 52,2; λ2 x 2β λ 2n2m C, x 2βn ǫ Γ2n2m2 2βn ǫ λ 2m2 x 2β ǫ 2 2m3 πγm2γm5/22β ǫ 2 F 3, ǫ 2β ;m2,m 5 ǫ,2 2 2β ; λ2 x 2β C, C,

6 8 Victor Nijimbere 3. Evaluation of the cosine integral Ci β,2βα, β, α > Theorem 3. Let β and α >, and let α 2βmǫ, where m is an integer m N and 2β < ǫ < 2β.. If ǫ 0, then cosλx β Ci β,2βα λx 2βα x 2β α λ 2β α m λ 2m x 2 m2 πγm3/2γm22β 2 F 3 where m α/2β. 2. If ǫ, then cosλx β λx 2βα λ x 2β α nm λ 2n2m x 2βn Γ2n2m3 2βn, 2β ;m 3 2,m2,2 x 2β 2β ; λ2 C, λ 2n2m 2β α m λ 2m ln x nm Γ2m3 Γ2n2m3 m λ 2m3 x 2β 2 2m5 πγm5/2γm3β 2 F 3,;m 52 x 2β,m3,2; λ2 C, where m α /2β. 3. Finally, if ǫ 2β,0 0,,2β, we have 3.5 x 2βn 2βn 3.6 cosλx β λx 2βα x 2β α λ 2β α m λ 2m x ǫ Γ2m3 ǫ where m α ǫ/2β. nm λ 2n2m x 2βn ǫ Γ2n2m3 2βn ǫ m λ 2m3 x 2β ǫ 2 2m πγm5/2γm32β ǫ 2 F 3, ǫ 2β ;m 5 ǫ x 2β,m3,2 2 2β ; λ2 C, 3.7 P r o o f. We proceed as in Theorem. We have cosλx β λx2βα λx 2βα nλxβ 2n dx 2n! λx 2βαdx n λ2n λ 2n! x2βn 2β α dx n λx 2βαdx n λ 2n2 λ 2n2! x2βn α dx m λx 2βαdx n λ2n 2n2! x2βn 2βm ǫ n λ2n dx 2n2! x2βn 2βm ǫ dx nm

7 Some non-elementary integrals of sine, cosine and exponential integrals type 9 λx 2βαdx m λx 2βαdx 2n2! x2βn m ǫ dx n λ2n λx 2βαdx nm nm λ 2m λx 2βαdx m Γ2m3 n nm nm λ 2n2m nm λ 2n2m n λ2n 2n2m2! x2βn ǫ dx 2n2m2! x2βn ǫ dx nm λ 2n2m λ 2n2m Γ2n2m3 x2βn ǫ dx Γ2n2m3 x2βn ǫ dx dx x ǫ λ 2n2m nm Γ2n2m3 x2βn ǫ dx 2n2! x2βn m ǫ dx λ 2n2m Γ2n2m3 x2βn ǫ dx λ 2m dx λx 2βαdx m Γ2m3 x ǫ nm λ 2n2m Γ2n2m3 x2βn ǫ dx nm λ 2n2m3 Γ2n2m5 x2βn2β ǫ dx x 2β α λ 2m dx λ 2β α m Γ2m3 x ǫ nm λ 2n2m x 2βn ǫ Γ2n2m3 2βn ǫ nm λ 2n2m3 x 2βn2β ǫ Γ2n2m5 2βn2β ǫ C x 2β α λ 2m dx λ 2β α m Γ2m3 x ǫ nm λ 2n2m x 2βn ǫ Γ2n2m3 2βn ǫ m λ 2m3 x 2β ǫ n ǫ/2β n λ 2 x 2β / n 2 2m C πγm5/2γm32β ǫ m5/2 n m3 n 2 ǫ/2β n n! x 2β α λ2m dx m λ 2β α Γ2m3 x ǫ nm λ 2n2m x 2βn ǫ Γ2n2m3 2βn ǫ m λ 2m3 x 2β ǫ 2 2m πγm5/2γm32β ǫ 2 F 3. For ǫ 0, we substitute ǫ 0 in 3.8, and hence, we obtain cosλx β λx2βα dx λx 2βα nm λ 2n2m, ǫ 2β ;m5 2,m3,2 ǫ x 2β 2β ; λ2 nm λ 2n2m Γ2n2m3 x2βn dx x 2β α Γ2n2m3 x2βn λ 2β α C. 3.8

8 50 Victor Nijimbere nm λ 2n2m x 2βn Γ2n2m3 2βn nm λ 2n2m x 2βn Γ2n2m3 2βn x 2β α λ 2β α nm λ 2n2m x 2βn Γ2n2m3 2βn m λ 2m x n /2β n λ 2 x 2β / n 2 m2 π2β Γm3/2Γm2 m3/2 n m2 n 2/2β n n! x 2β α λ 2β α m λ 2m x 2 m2 πγm3/2γm22β 2 F 3 which is 3.5, and where m α/2β. 2. For ǫ, we set ǫ in 3.8 and obtain nm λ 2n2m x 2βn Γ2n2m3 2βn, 2β ;m 3 2,m2,2 x 2β 2β ; λ2 C, cosλx β λx 2βα x 2β α λ 2β α m λ 2m Γ2m3 ln x nm λ 2n2m x 2βn Γ2n2m3 2βn m λ 2m3 x 2β 2 2m5 πγm5/2γm3β 2 F 3,;m 52 x 2β,m3,2; λ2 C, which is 3.6, and where m α /2β. 3. For ǫ 2β,0 0,,2β, 3.8 gives cosλx β λx 2βα x 2β α λ 2β α m λ 2m x ǫ Γ2m3 ǫ nm λ 2n2m x 2βn ǫ Γ2n2m3 2βn ǫ m λ 2m3 x 2β ǫ 2 2m πγm5/2γm32β ǫ 2 F 3, ǫ 2β ;m 5 ǫ x 2β,m3,2 2 2β ; λ2 C, which is 3.7, and where m α ǫ/2β. Example 2. In this example, we evaluate [ cosx/x 5] dx. We first observe that λ and β. We also have 5 2β α β 2βm ǫ, and so m and ǫ. Substituting λ, β, m and ǫ in 3.6 gives cosx x 5 x x 2 ln x 2 x2 720π 2 F 3,; 72,,2; x2 C. We can use the same procedure for the hyperbolic cosine integral, the results are stated in the next theorem. Its proof is similar to Theorem 3 s proof, we will omit it. Theorem. Let β and α >, and let α 2βmǫ, where m is an integer m N and 2β < ǫ < 2β.

9 Some non-elementary integrals of sine, cosine and exponential integrals type 5. If ǫ 0, then coshλx β λx 2βα λ x 2β α 2β α λ 2m x 2 m2 πγm3/2γm22β 2 F 3 where m α/2β. 2. If ǫ, then coshλx β λx 2βα λ x 2β α λ 2n2m x 2βn Γ2n2m3 2βn, 2β ;m 3 2,m2,2 2β ; λ2 x 2β Γ2m3 ln x λ 2n2m x 2βn Γ2n2m3 2βn C, 2β α λ2m λ 2m3 x 2β 2 2m5 πγm5/2γm3β 2 F 3,;m 52,m3,2; λ2 x 2β where m α /2β. 3. Finally, if ǫ 2β,0 0,,2β, we have cosλx β λx 2βα x 2β α λ 2β α m λ 2m x ǫ Γ2m3 ǫ where m α ǫ/2β. λ 2n2m λ 2m3 x 2β ǫ 2 2m πγm5/2γm32β ǫ 2 F 3, ǫ 2β ;m 5 ǫ,m3,2 2 2β ; λ2 x 2β C, C, x 2βn ǫ Γ2n2m3 2βn ǫ. Evaluation of the exponential integral Ei β,βα, β, α > Theorem 5. Let β and α >, and let α βmǫ, where m is an integer m N and β < ǫ < β.. If ǫ 0, then Ei β,βα λx βα x β α λ β α λ nm x βn Γnm2 βn λ m x Γm2β 2 F 2, β ;m2,2 β ;λxβ C, where m α/β. 2. If ǫ, then Ei β,βα where m α /β. λx βα x β α λ β α λ m Γm2 ln x λm x β Γm3β 2 F 2,;m3,2;λx β C, λ nm x βn Γnm2 βn.9.0

10 52 Victor Nijimbere 3. Finally, if ǫ β,0 0,,β, we have λx βα x β α λ β α λ m x ǫ Γm2 ǫ λ m x β ǫ Γm3β ǫ 2 F 2 where m α ǫ/β. λ nm, ǫ ǫ ;m3,2 β β ;λxβ P r o o f. We proceed as before. Then, we have λx βα λx β n λx βα n! λx βαdx λ n λ n λx βαdx λ n! xβn α dx m λ n λx βαdx n! xβn βm ǫ dx λx βαdx λx βαdx λx βαdx m λx βαdx λ n n! xβn m ǫ dx λ nm nm! xβn ǫ dx λ nm Γnm2 xβn ǫ dx λ m Γm2 λ nm Γnm2 xβn ǫ n dx x ǫ nm nm x βn ǫ Γnm2 βn ǫ C, λ n n! xβn β α dx λ n n! xβn βm ǫ dx λ n n! xβn m ǫ dx λ nm nm! xβn ǫ dx λ nm Γnm2 xβn ǫ dx λ nm Γnm2 xβn ǫ dx λ m dx λx βαdx Γm2 x ǫ λ nm λ nm Γnm2 xβn ǫ dx Γnm3 xβnβ ǫ dx x β α λ β α λ m dx Γm2 x ǫ λ nm x βn ǫ Γnm2 βn ǫ λ nm x βnβ ǫ Γnm3 βnβ ǫ C x β α λ β α λ m Γm2 λ m x β ǫ Γm3β ǫ x β α λ β α λ m Γm2 dx x ǫ λ nm n ǫ/β n m3 n 2 ǫ/β n dx x ǫ x βn ǫ Γnm2 βn ǫ λx β n C n! λ nm x βn ǫ Γnm2 βn ǫ..2

11 Some non-elementary integrals of sine, cosine and exponential integrals type 53 λ m x β ǫ Γm3β ǫ 2 F 2, ǫ ǫ ;m3,2 β β ;λxβ. For ǫ 0, we substitute ǫ 0 in.2, and hence, we obtain λx βα dx λx βα x β α λ β α λ nm Γnm2 xβn dx C. λ nm Γnm2 xβn dx λ nm x βn Γnm2 βn λ nm x βn Γnm2 βn x β α λ β α λ nm x βn Γnm2 βn λ m x n /β n λx β n x β α Γm2β m2 n 2/β n n! λ β α λ nm x βn Γnm2 βn which is.9, and where m α/β. 2. For ǫ, we set ǫ in.2 and obtain λ m x Γm2β 2 F 2, β ;m2,2 β ;λxβ C, λx βα x β α λ β α λ m Γm2 ln x λm x β Γm3β 2 F 2,;m3,2;λx β C, λ nm x βn Γnm2 βn which is.0, and where m α /β. 3. For ǫ β,0 0,,β,.2 gives λx βα x β α λ β α λ m x ǫ Γm2 ǫ λ m x β ǫ Γm3β ǫ 2 F 2 λ nm, ǫ ǫ ;m3,2 β β ;λxβ x βn ǫ Γnm2 βn ǫ C, which is., and where m α ǫ/β. Example 3. In this example, we evaluate e x2 /x dx. We first observe that λ and β 2. We also have βα ββmǫ, and so m and ǫ 0. Substituting λ, β, m and ǫ 0 in.9 gives e x 2 x 3 x 3 x x 2 F 2,2;3,3; x 2 C. Corollary. Let α > and let α mǫ, where m is an integer m N and < ǫ.

12 5 Victor Nijimbere. If ǫ 0 or, then Ei,α e λx λx α λαx α λ m Γm2 ln x λm x Γm3β 2 F 2,;m3,2;λxC, λ nm x n Γnm2 n where m α. 2. And if ǫ,0 0,, we have e λx λx α λαx α λ m x ǫ Γm2 ǫ λ nm x n ǫ Γnm2 n ǫ λm x 2 ǫ Γm32 ǫ 2 F 2,2 ǫ;m3,3 ǫ;λxc, where m α ǫ. P r o o f.. If ǫ 0 or implies α mǫ is an integer α N since m N. Morever, α mǫ implies β in Theorem 5. Therefore, we obtain by setting β in For ǫ,0 0,, we set β in. and obtain 2. Example. In this example, we evaluate e x /x 3.7 dx. We first observe that λ. We also have 3.7 α mǫ, and so m 2 and ǫ 0.7. Substituting λ,m 2 and ǫ 0.7 in 2 gives e x x 2.7 x x0.3.8 x.7.7 x 0.7. x F 2,.3;5,2.3; x C. 5. Conclusion Formulas for the non-elementary integrals Si β,α [sinλx β /λx α ]dx, β, α > β, and Ci β,α [cosλx β /λx α ]dx, β, α > 2β, were explicitly derived in terms of the hypergeometric function 2 F 3 see Theorems and 2. Once derived, formulas for the hyperbolic sine and hyperbolic cosine integrals were deduced from those of the sine and cosine integrals see Theorems 2 and. On the other hand, the exponential integral Ei β,α /x α dx, β, α > β was expressed in terms of the hypergeometric function 2 F 2 see Theorem 5 and Corollary. Beside, illustrative examples were given. Therefore, their corresponding definite integrals can now be evaluated using the FTC rather than using numerical integration. REFERENCES. Abramowitz M., Stegun I.A. Handbook of mathematical functions with formulas, graphs and mathematical tables. National Bureau of Standards, p. 2. Chiccoli C., Lorenzutta S., Maino G. Concerning some integrals of the generalized exponential-integral function. Computers Math. Applic., 992. Vol. 23, no., P DOI: 0.06/ P

13 Some non-elementary integrals of sine, cosine and exponential integrals type Chen X. Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks. Ann. Probab., 200. Vol. 32, no.. P DOI: 0.2/ Marchisotto E.A., Zakeri G.-A. An invitation to integration in finite terms. College Math. J., 99. Vol. 25, no.. P DOI: / Nijimbere V. Evaluation of the non-elementary integral e λxα dx, α 2, and other related integrals. Ural Math. J., 207. Vol. 3, no. 2. P DOI: /umj Nijimbere V. Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals: Part I Ural Math. J., 208. Accepted for publication. 7. NIST Digital Library of Mathematical Functions Rahman M. Applications of Fourier transforms to generalized functions. Witt Press, p. 9. Rosenlicht M. Integration in finite terms. Amer. Math. Monthly, 972. Vol. 79, no. 9. P DOI: / Shore S.N. Blue sky and hot piles: the evolution of radiative transfer theory from atmospheres to nuclear reactors. Historia Mathematica, Vol. 29, no. 2. P DOI: 0.006/hmat