Introductions to ExpIntegralE Introduction to the exponential integrals General The exponential-type integrals have a long history. After the early developments of differential calculus, mathemati- cians tried to evaluate integrals containing simple elementary functions, especially integrals that often appeared during investigations of physical problems. Despite the relatively simple form of the integrands, some of these integrals could not be evaluated through known functions. Examples of integrals that could not be evaluated in known functions are: 1 ât log t sin t à ât tH L cosH tL à ât t ãt H L à ât. t L. Euler (1768) introduced the first integral shown in the preceding list. Later L. Mascheroni (1790, 1819) used it à and introduced the second and third integrals, and A. M. Legendre (1811) introduced the last integral shown. T. Caluso (1805) used the first integral in an article and J. von Soldner (1809) introduced its notation through symbol li. F. W. Bessel (1812) used the second and third integrals. C. A. Bretschneider (1843) not only used the second and third integrals, but also introduced similar integrals for the hyperbolic functions: sinh t cosh t ât ât. t t O. SchlöH L milch (1846)H L and F. Arndt (1847) widely used such integrals containing the exponential and trigonometric à à functions. For the exponential, sine, and cosine integrals, J. W. L. Glaisher (1870) introduced the notations Ei, Si, and Ci. H. Amstein (1895) introduced the branch cut for the logarithmic integral with a complex argument. N. Nielsen (1904) used the notations Si and Ci for corresponding integrals. Different notations are used for the previous definite integrals by various authors when they are integrated from 0 to z or from z to ¥. Definitions of exponential integrals The exponential integral En z , exponential integral Ei z , logarithmic integral li z , sine integral Si z , hyperbolic sine integral Shi z , cosine integral Ci z , and hyperbolic cosine integral Chi z are defined as the following definite integrals, including the Euler gamma constant g 0.577216¼: H L H L H L H L H L H L H L http://functions.wolfram.com 2 ¥ ã-z t En z ât ; Re z > 0 n 1 t z ãt - 1 1 1 EiHzL à ât+ HlogL z - log + ý 0 t 2 z z 1 li zH L à ât H L 0 log t z sin t SiH zL à ât 0 tH L z sinhH L t ShiH Lz à ât 0 t z cos t H-L 1 Ci zH L à ât + log z + ý 0 t z coshH L t - 1 ChiH Lz à ât + logH L z + ý. 0 t The previous integralsH L are all interrelated and are called exponential integrals. H L à H L Instead of the above classical definitions through definite integrals, equivalent definitions through infinite series can be used, for example, the exponential integral Ei z can be defined by the following formula (see the following sections for the corresponding series for the other integrals): 1 1 ¥ zk H L Ei z log z - log + + ý. 2 z k=1 k k! A quickH L lookH Lat the exponentialâ integrals Here is a quick look at the graphics for the exponential integrals along the real axis. 4 2 E-3 x f 0 E-1 x H L -2 E1 x H L -4 E3 x H L -4 -2 0 2 4 E x x 5 H L H L http://functions.wolfram.com 3 4 2 Si x f 0 Ci x H L -2 Shi x H L -4 Chi x H L -7.5 -5 -2.5 0 2.5 5 7.5 x H L 4 2 Ei x f 0 li x H L -2 H L -4 -7.5 -5 -2.5 0 2.5 5 7.5 x Connections within the group of exponential integrals and with other function groups Representations through more general functions The exponential integrals En z , Ei z , li z , Si z , Shi z , Ci z , and Chi z are the particular cases of the more general hypergeometric and Meijer G functions. For example, they can be representedH L H L throughH L hypergeometricH L H L H functionsL pHFqL or the Tricomi confluent hypergeomet- ric function U: 1 n-1 En z z G 1 - n - 1F1 1 - n; 2 - n; -z 1 - n E z zn-1 ã-z U n, n, z nH L H L H L 1 1 Ei z z 2F2 1, 1; 2, 2; z + log z - log + ý H L H L 2 z 1 1 EiHzL -ãz UH 1, 1, -z -Llog -z +H L log z - log 2 z 1 1 li zH L log z 2HF2 1, 1; L2, 2; logH zL + logH logL z - log + ý 2 log z 1 3 3 z2 H L H L H H LL H H LL Si z z 1F2 ; , ; - 2 2 2 4 H L 2 H L 1 3 3 z Shi z z 1F2 ; , ; 2 2 2 4 2 2 H L z 3 z Ci z - 2F3 1, 1; 2, 2, ; - + log z + ý 4 2 4 H L H L http://functions.wolfram.com 4 z2 3 z2 Chi z 2F3 1, 1; 2, 2, ; + log z + ý. 4 2 4 RepresentationsH L of the exponential integralsH L En z and Ei z , the sine and cosine integrals Si z and Ci z , and the hyperbolic sine and cosine integrals Shi z and Chi z through classical Meijer G functions are rather simple: 2,0 n H L H L H L H L En z G1,2 z n - 1, 0 H L H L 1,2 1, 1 1 1 EiHzL -G2,3 -z + ý - log - log z 1, 0, 0 2 z 2 1 H L p 1,1 z 2 H L Si z z G1,3 4 4 0, - 1 , - 1 2 2 H L p z2 1 1 Ci z - G2,0 - log z2 - 2 log z 1,3 0, 0, 1 2 4 2 2 2 1 H L p 1,1 z 2 I I M H LM Shi z z G1,3 - 4 4 0, - 1 , - 1 2 2 H L p z2 1 1 Chi z - G2,0 - - log -z2 - 2 log z . 1,3 0, 0, 1 2 4 2 2 HereH Lý is the Euler gamma constant ýI 0.577216I M ¼H andLM the complicated-looking expression containing the two logarithm simplifies piecewise: 1 1 - log - log z log z ; z Ï -¥, 0 2 z 1 1 - log - logHzL logHzL - p ä ;8z Î -¥< , 0 . 2 z But the last four formulas that contain the Meijer G function can be simplified further by changing the classical H L H L 8 < Meijer G functions to the generalized one. These formulas do not include factors z and terms 1 log z2 - 2 log z : 2 p z 1 1 Si z G1,1 , 1,3 1 , 0, 0 I I M H LM 2 2 2 2 H L p z 1 1 Ci z - G2,0 , 1,3 0, 0, 1 2 2 2 2 H L ä p ä z 1 1 Shi z - G1,1 , 1,3 1 , 0, 0 2 2 2 2 1 H L 1 z 1 , 1 3 2 2,0 2 Chi z - p G2,4 , . 2 2 2 0, 0, 1 , 1 2 2 H L http://functions.wolfram.com 5 The corresponding representations of the logarithmic integral li z through the classical Meijer G function is more complicated and includes composition of the G function and a logarithmic function: 1 1 1,2 1, 1 H L li z ý - log - log log z - G2,3 -log z . 2 log z 1, 0, 0 All six exponential integrals of one variable are the particular cases of the incomplete gamma function: H L H H LL H L H L 1 1 Ei z -G 0, -z + log z - log - log -z 2 z ä SiHzL GH 0, -Lä z - G 0,HäLz + log -ä z - logH äLz 2 1 ShiH Lz H GH 0, z -L G 0H, -z -L log -H z +Llog z H LL 2 1 Ci zH L logHz H- L G 0H, -ä z L+ G 0,Hä zL + logH-LäLz + log ä z 2 1 ChiH Lz - H L G 0, H-zH + G 0L, z +Hlog -L z - logH z L H LL 2 1 1 li z HL -G 0, H-logH z L+ H logLlog z H - logL H LL - log -log z . 2 log z Representations through related equivalent functions H L H H LL H H LL H H LL H L The exponential integral En z can be represented through the incomplete gamma function or the regularized incomplete gamma function: n-1 En z z G 1 - n, z H L n-1 En z z G 1 - n Q 1 - n, z . H L H L Relations to inverse functions H L H L H L The exponential integral En z is connected with the inverse of the regularized incomplete gamma function Q-1 a, z by the following formula: -1 -1 H nL-1 En Q 1 - n, z Q 1 - n, z G 1 - n z. H L Representations through other exponential integrals I H LM H L H L The exponential integrals En z , Ei z , li z , Si z , Shi z , Ci z , and Chi z are interconnected through the follow- ing formulas: 1 H L 1 H L H L H L H L H L H L Ei z -E1 -z + log z - log - log -z 2 z Ei z li ãz ; -p < Im z £ p H L H L H L H L Ei log z li z H L H L H L H H LL H L http://functions.wolfram.com 6 1 1 Ei z Ci ä z - ä Si ä z - log - log z - log ä z 2 z 1 1 EiHzL ChiH zL + ShiHz -L log + log zH L H L 2 z 1 1 li zH L -E1 H-Llog z H+L log log z - logH L - log -log z 2 log z li z Ei log z H L H H LL H H LL H H LL H L 1 1 li z Ci ä log z - ä Si ä log z - log - log log z - log ä log z H L H H LL 2 log z 1 1 liHzL ChiH logHzLL + ShiHlog zH L-L log + log logH zH LL H H LL 2 log zH L ä SiH zL HE1 -H äLLz - E1Hä z H+LLlog -ä z - log ä z H H LL 2 H L ä ä ä SiHzL H2 HEi -Lä z - EiH äLz + logH L - logH -LL - log -ä z + log ä z 4 z z 1 p SiHzL liH ãäHz - Lli ã-ä Hz -LL sgn Re z ; Re z < pH L H L 2 ä 2 Si z -ä Shi ä z H L I I M I MM H H LL H L¤ 1 Shi z E1 z - E1 -z - log -z + log z H L 2 H L 1 1 1 ShiHzL H2 HEiL z - EiH -Lz + logH L - logH LL- + log -z - log z 4 z z 1 ä p ShiHzL li Hãz H-Lli ã-Hz -LL sgn Im z ; Im z < pH L H L 2 2 Shi z -ä Si ä z H L H H L H LL H H LL H L¤ 1 Ci z - E1 -ä z + E1 ä z + log -ä z + log ä z + log z H L 2 H L 1 ä ä CiHzL 2H EiH -ä zL + EiHä zL + logH L+ log H- LL - logH-Lä z - log ä z + log z 4 z z 1 p ä CiHzL li Hã-äHz + liL ãä z H +LL sgn Im z 1 - sgn Re zH ;L Re z H <Lp H L 2 2 Ci z Chi ä z - log ä z + log z H L I I M I MM H H LL H H H LLL H L¤ 1 Chi z - E1 -z + E1 z + log -z - log z H L 2H L H L H L 1 1 1 ChiHzL 2H EiH -zL + EiHzL + logH L + logH -LL - log -z + 3 log z 4 z z H L H H L H LL H L H L http://functions.wolfram.com 7 1 p ä 1 1 Chi z li ã-z + li ãz + sgn Im z + log + log z ; Im z < p 2 2 2 z Chi z Ci ä z + log z - log ä z .