Integration in Finite Terms with Special Functions: Error Functions, Logarithmic Integrals and Polylogarithmic Integrals. Yashpreet Kaur, V. Ravi Srinivasan∗ IISER Mohali, INDIA March 22, 2019 YK-VRS Integration with special functions 1 / 31 Introduction Notations and terminologies Throughout, a field always means a field of characteristic zero. Differential fields: A field F with an additive map 0 : F ! F that satisfies the Leibnitz rule, i.e (fg)0 = fg0 + f 0g for all f; g 2 F . 0 The kernel of the map is denoted by CF , called the field of constants. Differential field extension: A differential field E is said to differential field extension of F if E is a field extension of F and the derivation map of E restricted to F coincides with the derivation map of F . YK-VRS Integration with special functions 2 / 31 Introduction Let E be a differential field extension of F having the same field of constants as F . Problem When an element α 2 F admits an antiderivative in E? YK-VRS Integration with special functions 3 / 31 (i) θi is algebraic over Fi−1: 0 0 u (ii) θi = u θi for u 2 Fi−1 (i.e. θi = e ; called an exponential of u). 0 0 (iii) θi = u =u for u 2 Fi−1 (i.e. θi = log(u); called a logarithm of u). 0 0 (iv) θi = u = log u, where u; log u 2 Fi−1 (i.e. θi = `i(u); called logarithmic integral of u). 0 0 −u2 −u2 (v) θi = u e , where u; e 2 Fi−1 (i.e. θi = erf (u); called error function of u). 0 0 (vi) θi = − log(1 − u)u =u, where u; log(1 − u) 2 Fi−1 (i.e. θi = `2(u); called dilogarithmic integral of u). 0 0 (vii) θi = `2(u)u =u, where u; `2(u) 2 Fi−1 (i.e. θi = `3(u); called trilogarithmic integral of u). Introduction We will be working with differential field extensions of the form E = F (θ1; : : : ; θn), F0 := F , Fi = Fi−1(θi), CE = CF such that for each i, one of the following holds: YK-VRS Integration with special functions 4 / 31 0 0 (iv) θi = u = log u, where u; log u 2 Fi−1 (i.e. θi = `i(u); called logarithmic integral of u). 0 0 −u2 −u2 (v) θi = u e , where u; e 2 Fi−1 (i.e. θi = erf (u); called error function of u). 0 0 (vi) θi = − log(1 − u)u =u, where u; log(1 − u) 2 Fi−1 (i.e. θi = `2(u); called dilogarithmic integral of u). 0 0 (vii) θi = `2(u)u =u, where u; `2(u) 2 Fi−1 (i.e. θi = `3(u); called trilogarithmic integral of u). Introduction We will be working with differential field extensions of the form E = F (θ1; : : : ; θn), F0 := F , Fi = Fi−1(θi), CE = CF such that for each i, one of the following holds: (i) θi is algebraic over Fi−1: 0 0 u (ii) θi = u θi for u 2 Fi−1 (i.e. θi = e ; called an exponential of u). 0 0 (iii) θi = u =u for u 2 Fi−1 (i.e. θi = log(u); called a logarithm of u). YK-VRS Integration with special functions 4 / 31 0 0 −u2 −u2 (v) θi = u e , where u; e 2 Fi−1 (i.e. θi = erf (u); called error function of u). 0 0 (vi) θi = − log(1 − u)u =u, where u; log(1 − u) 2 Fi−1 (i.e. θi = `2(u); called dilogarithmic integral of u). 0 0 (vii) θi = `2(u)u =u, where u; `2(u) 2 Fi−1 (i.e. θi = `3(u); called trilogarithmic integral of u). Introduction We will be working with differential field extensions of the form E = F (θ1; : : : ; θn), F0 := F , Fi = Fi−1(θi), CE = CF such that for each i, one of the following holds: (i) θi is algebraic over Fi−1: 0 0 u (ii) θi = u θi for u 2 Fi−1 (i.e. θi = e ; called an exponential of u). 0 0 (iii) θi = u =u for u 2 Fi−1 (i.e. θi = log(u); called a logarithm of u). 0 0 (iv) θi = u = log u, where u; log u 2 Fi−1 (i.e. θi = `i(u); called logarithmic integral of u). YK-VRS Integration with special functions 4 / 31 0 0 (vi) θi = − log(1 − u)u =u, where u; log(1 − u) 2 Fi−1 (i.e. θi = `2(u); called dilogarithmic integral of u). 0 0 (vii) θi = `2(u)u =u, where u; `2(u) 2 Fi−1 (i.e. θi = `3(u); called trilogarithmic integral of u). Introduction We will be working with differential field extensions of the form E = F (θ1; : : : ; θn), F0 := F , Fi = Fi−1(θi), CE = CF such that for each i, one of the following holds: (i) θi is algebraic over Fi−1: 0 0 u (ii) θi = u θi for u 2 Fi−1 (i.e. θi = e ; called an exponential of u). 0 0 (iii) θi = u =u for u 2 Fi−1 (i.e. θi = log(u); called a logarithm of u). 0 0 (iv) θi = u = log u, where u; log u 2 Fi−1 (i.e. θi = `i(u); called logarithmic integral of u). 0 0 −u2 −u2 (v) θi = u e , where u; e 2 Fi−1 (i.e. θi = erf (u); called error function of u). YK-VRS Integration with special functions 4 / 31 0 0 (vii) θi = `2(u)u =u, where u; `2(u) 2 Fi−1 (i.e. θi = `3(u); called trilogarithmic integral of u). Introduction We will be working with differential field extensions of the form E = F (θ1; : : : ; θn), F0 := F , Fi = Fi−1(θi), CE = CF such that for each i, one of the following holds: (i) θi is algebraic over Fi−1: 0 0 u (ii) θi = u θi for u 2 Fi−1 (i.e. θi = e ; called an exponential of u). 0 0 (iii) θi = u =u for u 2 Fi−1 (i.e. θi = log(u); called a logarithm of u). 0 0 (iv) θi = u = log u, where u; log u 2 Fi−1 (i.e. θi = `i(u); called logarithmic integral of u). 0 0 −u2 −u2 (v) θi = u e , where u; e 2 Fi−1 (i.e. θi = erf (u); called error function of u). 0 0 (vi) θi = − log(1 − u)u =u, where u; log(1 − u) 2 Fi−1 (i.e. θi = `2(u); called dilogarithmic integral of u). YK-VRS Integration with special functions 4 / 31 Introduction We will be working with differential field extensions of the form E = F (θ1; : : : ; θn), F0 := F , Fi = Fi−1(θi), CE = CF such that for each i, one of the following holds: (i) θi is algebraic over Fi−1: 0 0 u (ii) θi = u θi for u 2 Fi−1 (i.e. θi = e ; called an exponential of u). 0 0 (iii) θi = u =u for u 2 Fi−1 (i.e. θi = log(u); called a logarithm of u). 0 0 (iv) θi = u = log u, where u; log u 2 Fi−1 (i.e. θi = `i(u); called logarithmic integral of u). 0 0 −u2 −u2 (v) θi = u e , where u; e 2 Fi−1 (i.e. θi = erf (u); called error function of u). 0 0 (vi) θi = − log(1 − u)u =u, where u; log(1 − u) 2 Fi−1 (i.e. θi = `2(u); called dilogarithmic integral of u). 0 0 (vii) θi = `2(u)u =u, where u; `2(u) 2 Fi−1 (i.e. θi = `3(u); called trilogarithmic integral of u). YK-VRS Integration with special functions 4 / 31 Introduction Elementary Extensions A differential field extension E = F (θ1; : : : ; θn) of F is called an elementary extension if each θi is either algebraic, exponential or logarithmic over Fi−1: Elements of an elementary extension field are called elementary functions . YK-VRS Integration with special functions 5 / 31 M. Rosenlicht (1968) was the first to give a purely algebraic solution to the problem. History (elementary functions) The problem of integration in finite terms for elementary functions was considered by J. Liouville (1834-35) and by J.F. Ritt (1948). YK-VRS Integration with special functions 6 / 31 History (elementary functions) The problem of integration in finite terms for elementary functions was considered by J. Liouville (1834-35) and by J.F. Ritt (1948). M. Rosenlicht (1968) was the first to give a purely algebraic solution to the problem. YK-VRS Integration with special functions 6 / 31 Error functions and logarithmic integrals, G. Cherry (1985-86). EL −elementary functions, M. Singer, B. Saunders andB. Caviness (1985). Polylogarithmic integrals do not belong to this class. Dilogarithmic integrals, J. Baddoura (2006). History (Special Functions) J. Moses (1969). YK-VRS Integration with special functions 7 / 31 EL −elementary functions, M. Singer, B. Saunders andB. Caviness (1985). Polylogarithmic integrals do not belong to this class. Dilogarithmic integrals, J. Baddoura (2006). History (Special Functions) J. Moses (1969). Error functions and logarithmic integrals, G. Cherry (1985-86). YK-VRS Integration with special functions 7 / 31 Dilogarithmic integrals, J. Baddoura (2006). History (Special Functions) J. Moses (1969). Error functions and logarithmic integrals, G. Cherry (1985-86). EL −elementary functions, M. Singer, B. Saunders andB. Caviness (1985). Polylogarithmic integrals do not belong to this class. YK-VRS Integration with special functions 7 / 31 History (Special Functions) J.