Integration in Finite Terms with Special Functions: Error Functions, Logarithmic Integrals and Polylogarithmic Integrals

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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 ∈ 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 ∈ 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 ∈ Fi−1 (i.e. θi = e , called an exponential of u). 0 0 (iii) θi = u /u for u ∈ Fi−1 (i.e. θi = log(u), called a logarithm of u). 0 0 (iv) θi = u / log u, where u, log u ∈ Fi−1 (i.e. θi = `i(u), called logarithmic integral of u).

0 0 −u2 −u2 (v) θi = u e , where u, e ∈ 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) ∈ Fi−1 (i.e. θi = `2(u), called dilogarithmic integral of u).

0 0 (vii) θi = `2(u)u /u, where u, `2(u) ∈ 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 ∈ Fi−1 (i.e. θi = `i(u), called logarithmic integral of u).

0 0 −u2 −u2 (v) θi = u e , where u, e ∈ 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) ∈ Fi−1 (i.e. θi = `2(u), called dilogarithmic integral of u).

0 0 (vii) θi = `2(u)u /u, where u, `2(u) ∈ 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 ∈ Fi−1 (i.e. θi = e , called an exponential of u). 0 0 (iii) θi = u /u for u ∈ 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 ∈ 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) ∈ Fi−1 (i.e. θi = `2(u), called dilogarithmic integral of u).

0 0 (vii) θi = `2(u)u /u, where u, `2(u) ∈ 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.

YK-VRS Integration with special functions 4 / 31 0 0 (vi) θi = − log(1 − u)u /u, where u, log(1 − u) ∈ Fi−1 (i.e. θi = `2(u), called dilogarithmic integral of u).

0 0 (vii) θi = `2(u)u /u, where u, `2(u) ∈ 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 −u2 −u2 (v) θi = u e , where u, e ∈ 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) ∈ 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 −u2 −u2 (v) θi = u e , where u, e ∈ 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) ∈ 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 −u2 −u2 (v) θi = u e , where u, e ∈ 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) ∈ Fi−1 (i.e. θi = `2(u), called dilogarithmic integral of u).

0 0 (vii) θi = `2(u)u /u, where u, `2(u) ∈ 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)

YK-VRS Integration with special functions 7 / 31 We obtain a simpler proof of Baddoura’s theorem which neither assumes that F is a liouvillian extension of CF nor that CF is an algebraically closed field. Our results contain both necessary and sufficient conditions and therefore, these results will help in formulating algorithms for integration in finite terms with special functions.

Highlights of our work

We extend Liouville’s theorem to include error functions, logarithmic integrals, dilogarithmic and trilogarithmic integrals in its ﬁeld of deﬁnition.

YK-VRS Integration with special functions 8 / 31 Our results contain both necessary and suﬃcient conditions and therefore, these results will help in formulating algorithms for integration in ﬁnite terms with special functions.

Highlights of our work

YK-VRS Integration with special functions 8 / 31 Highlights of our work

We extend Liouville’s theorem to include error functions, logarithmic integrals, dilogarithmic and trilogarithmic integrals in its field of definition. We obtain a simpler proof of Baddoura’s theorem which neither assumes that F is a liouvillian extension of CF nor that CF is an algebraically closed field. Our results contain both necessary and sufficient conditions and therefore, these results will help in formulating algorithms for integration in finite terms with special functions.

YK-VRS Integration with special functions 8 / 31 Liouville’s Theorem

Theorem (Rosenlicht, 1968)

Let E ⊃ F be an elementary ﬁeld extension of F with CE = CF . If there is an element u ∈ E with u0 ∈ F then there are Q−linearly independent constants c1, . . . , cn and elements g1, . . . , gn, w in F such that

n X g0 u0 = c i + w0. i g i=1 i

YK-VRS Integration with special functions 9 / 31 A differential field extension E = F (θ1, . . . , θn) is called a transcendental DEL-extension if CE = CF and each θi is transcendental over Fi−1 and satisfies one of the following:

(i) θi is an exponential over Fi−1.

(ii) θi is a logarithm over Fi−1.

(iii) θi is an error function over Fi−1.

(iv) θi is a logarithmic integral over Fi−1.

(v) θi is a dilogarithmic integral over Fi−1.

Dilogarithmic Integrals and DEL−Expressions

Recall that dilogarithmic integral of an element g ∈ F − {0, 1} is deﬁned as Z g0 l (g) = − log(1 − g). 2 g

YK-VRS Integration with special functions 10 / 31 Dilogarithmic Integrals and DEL−Expressions

Recall that dilogarithmic integral of an element g ∈ F − {0, 1} is deﬁned as Z g0 l (g) = − log(1 − g). 2 g

A differential field extension E = F (θ1, . . . , θn) is called a transcendental DEL-extension if CE = CF and each θi is transcendental over Fi−1 and satisfies one of the following:

(i) θi is an exponential over Fi−1.

(ii) θi is a logarithm over Fi−1.

(iii) θi is an error function over Fi−1.

(iv) θi is a logarithmic integral over Fi−1.

(v) θi is a dilogarithmic integral over Fi−1.

YK-VRS Integration with special functions 10 / 31 A DEL−expression is called (a)a special DEL−expression if for each i, 0 0 ri = ci(1 − gi) /(1 − gi) for some constant ci. (b)a D−expression if it is special and for all j, k, aj = bk = 0.

DEL−Extensions and DEL−Expressions

We say that v ∈ F admits a DEL−expression over F if there are some ﬁnite indexing sets I, J, K and elements −v2 w, ri, gi, uj, log(uj), vk, e k in F and constants aj, bk for all i, j, k in I, J, K respectively such that

0 0 X gi X uj X 0 −v2 0 v = ri + aj + bkvke k + w , gi log(uj) i∈I j∈J k∈K

where for each i, there is an integer ni such that 0 Pni 0 ri = l=1 cilhil/hil for some constants cil and elements hil ∈ F.

YK-VRS Integration with special functions 11 / 31 (b)a D−expression if it is special and for all j, k, aj = bk = 0.

DEL−Extensions and DEL−Expressions

0 0 X gi X uj X 0 −v2 0 v = ri + aj + bkvke k + w , gi log(uj) i∈I j∈J k∈K

where for each i, there is an integer ni such that 0 Pni 0 ri = l=1 cilhil/hil for some constants cil and elements hil ∈ F. A DEL−expression is called (a)a special DEL−expression if for each i, 0 0 ri = ci(1 − gi) /(1 − gi) for some constant ci.

YK-VRS Integration with special functions 11 / 31 DEL−Extensions and DEL−Expressions

0 0 X gi X uj X 0 −v2 0 v = ri + aj + bkvke k + w , gi log(uj) i∈I j∈J k∈K

YK-VRS Integration with special functions 11 / 31 A transcendental DEL−extension will be called transcendental dilogarithmic-elementary extension of F if for each i, θi is either an exponential or logarithm or dilogarithmic integral over Fi−1.

DEL−Extensions and DEL−Expressions

A diﬀerential ﬁeld extension E of F will be called a logarithmic extension of F if CE = CF and there are elements h1, . . . , hm in F such that E = F (log h1,..., log hm).

YK-VRS Integration with special functions 12 / 31 DEL−Extensions and DEL−Expressions

A diﬀerential ﬁeld extension E of F will be called a logarithmic extension of F if CE = CF and there are elements h1, . . . , hm in F such that E = F (log h1,..., log hm). A transcendental DEL−extension will be called transcendental dilogarithmic-elementary extension of F if for each i, θi is either an exponential or logarithm or dilogarithmic integral over Fi−1.

YK-VRS Integration with special functions 12 / 31 Furthermore, if E is a transcendental dilogarithmic-extension of F then u0 admits a D−expression over some logarithmic extension of F.

The theorem follows from the next lemma.

Integration in Finite Terms: DEL−Extensions

YK-VRS Integration with special functions 13 / 31 The theorem follows from the next lemma.

Integration in Finite Terms: DEL−Extensions

Theorem (YK-VRS, J. Symb. Comp, 94 (2019) 210-233.) Let E ⊃ F be a transcendental DEL−extension of F. Suppose that there is an element u in E with u0 in F then u0 admits a special DEL−expression over some logarithmic extension of F. Furthermore, if E is a transcendental dilogarithmic-extension of F then u0 admits a D−expression over some logarithmic extension of F.

YK-VRS Integration with special functions 13 / 31 Integration in Finite Terms: DEL−Extensions

The theorem follows from the next lemma.

YK-VRS Integration with special functions 13 / 31 Integration in Finite Terms: DEL−Extensions

Lemma (YK-VRS, J. Symb. Comp, 94 (2019) 210-233.) Let F (θ) ⊃ F be a transcendental DEL−extension of F. If v ∈ F admits a special DEL−expression over the differential field F (θ)(log y1,..., log yn), where each yi ∈ F (θ), having the same field of constants as F then there is a differential field M = F (log h1,..., log hm, θ), where each hi ∈ F, having the same field of constants as F such that each v admits a special DEL−expression over M.

YK-VRS Integration with special functions 14 / 31 E is a dilogarithmic-elementary extension of F. 0 u := `2(1 − z(z + 1)) + `2(1 − z) + v0 ∈ E,u = v ∈ F. Over the ﬁeld F (log z), for element 2 2 v0 = −(1/2) log (z) + log z(z − 1)(z + z − 1) log(z) + w in F (log z), we can rewrite v as

(1 − z(z + 1))0 (1 − z)0 u0 = − log(z(z + 1)) − log z + v0 1 − z(z + 1) 1 − z 0

which is a D-expression over F (log z).

Example

YK-VRS Integration with special functions 15 / 31 Example

2 Let F = C z, log(z + 1), log(z(z − 1)(z + z − 1)) and E = F (log z, `2(1 − z), `2(1 − z(z + 1)) be differential fields with 0 0 (1−z(z+1)) the derivation := d/dx. Let v = − log(z + 1) 1−z(z+1) + 2 z0 0 log z(z − 1)(z + z − 1) z + w ∈ F, where w ∈ F is arbitrary. Then we have the following: E is a dilogarithmic-elementary extension of F. 0 u := `2(1 − z(z + 1)) + `2(1 − z) + v0 ∈ E,u = v ∈ F. Over the field F (log z), for element 2 2 v0 = −(1/2) log (z) + log z(z − 1)(z + z − 1) log(z) + w in F (log z), we can rewrite v as

(1 − z(z + 1))0 (1 − z)0 u0 = − log(z(z + 1)) − log z + v0 1 − z(z + 1) 1 − z 0

which is a D-expression over F (log z).

YK-VRS Integration with special functions 15 / 31 Example

u0 cannot be written as a D−expression over F.

YK-VRS Integration with special functions 16 / 31 Integration in Finite Terms: DEL−Extensions

Theorem (YK, PhD Thesis, IISERM) Let E ⊃ F be a transcendental DEL−extension of F. Then there is an element u ∈ E with u0 ∈ F, if and only if u0 satisﬁes the following DEL−expression over F :

0 0 u0 0 X gi X hl X j X 0 −v2 0 u = ri + sl + aj + bkvke k + w , gi hl log(uj) i∈I l∈L j∈J k∈K

where for each i, p ∈ I and l, t ∈ L, there are constants aip, blt, cip and dil with cii 6= 0 such that

0 0 0 0 X (1 − gp) X gp X hl ri = cip + aip + dil and 1 − gp gp hl p∈I p∈I l∈L

0 0 0 X gi X ht sl = dil + blt . gi ht i∈I t∈L That is, u0 admits a DEL−YK-VRSexpressionIntegration over withF. special functions 17 / 31 Example (converse of the above theorem)

Let the differential field F := C(x, ln(x), ln(1 − x) + 5 ln(1 + x)) with derivation 0 := d/dx. Note that 1 1 v := (ln(1 − x) + 5 ln(1 + x)) + 5 ln(x) + w0, x 1 + x where w ∈ F is a DEL−expression over F of the desired form. Over the differential field F (log(1 + x)), we shall rewrite v as

1 1 1 0 v := ln(1 − x) + 5 ln(1 + x) + ln(x) + w0. x x 1 + x

Thus the dilogarithmic-elementary extension ﬁeld F (ln(1+x), `2(x)) contains an antiderivative of v, namely,

−`2(x) + 5 ln(1 + x) ln(x) + w.

YK-VRS Integration with special functions 18 / 31 Generalisation of Baddoura’s Theorem

The element D(g) := l2(g) + (1/2) log(g) log(1 − g) is called the Bloch-Wigner Spence function of g and its derivative is

1 g0 1 (1 − g)0 D(g)0 = − log(1 − g) + log(g). 2 g 2 (1 − g)

YK-VRS Integration with special functions 19 / 31 Generalisation of Baddoura’s Theorem

Our theorem concerning dilogarithmic integrals leads to a simpler version of Baddoura’s theorem (2006):

Theorem (YK-VRS, J. Symb. Comp, 94 (2019) 210-233.) Let E ⊃ F be a transcendental dilogarithmic-elementary extension of F. Suppose that there is an element u ∈ E with u0 ∈ F, then

m n X X u = cjD(gj) + fi log(hi) + w, j=1 i=1

where each fi, hi, gj, w ∈ F, cj are constants and log(hi) and D(gj) belong to some dilogarithmic-elementary extension of F.

YK-VRS Integration with special functions 20 / 31 0 For u := `2(1 − z(z + 1)) + `2(1 − z) + v0 ∈ E,u = v ∈ F , the Bloch Wigner Spence function representation is given by

u = D(1 − z(z + 1)) + D(1 − z) 1 1 + log z log z(z − 1)(z2 + z − 1) + c log z 2 2 1 − log(z + 1) log(1 − z(z + 1)) + v . 2 0

Example

YK-VRS Integration with special functions 21 / 31 Example

2 Let F = C z, log(z + 1), log(z(z − 1)(z + z − 1)) and E = F (log z, `2(1 − z), `2(1 − z(z + 1)) be diﬀerential ﬁelds with 0 0 (1−z(z+1)) the derivation := d/dx. Let v = − log(z + 1) 1−z(z+1) + 2 z0 0 log z(z − 1)(z + z − 1) z + w ∈ F, where w ∈ F is arbitrary. 0 For u := `2(1 − z(z + 1)) + `2(1 − z) + v0 ∈ E,u = v ∈ F , the Bloch Wigner Spence function representation is given by

u = D(1 − z(z + 1)) + D(1 − z) 1 1 + log z log z(z − 1)(z2 + z − 1) + c log z 2 2 1 − log(z + 1) log(1 − z(z + 1)) + v . 2 0

YK-VRS Integration with special functions 21 / 31 A differential field extension E = F (θ1, . . . , θn) is called a transcendental TEL-extension if CE = CF and each θi is transcendental over Fi−1 and satisfies one of the following:

(i) θi is an exponential over Fi−1.

(ii) θi is a logarithm over Fi−1.

(iii) θi is an error function over Fi−1.

(iv) θi is a logarithmic integral over Fi−1.

(v) θi is a dilogarithmic integral over Fi−1.

(vi) θi is a trilogarithmic integral over Fi−1.

Trilogarithmic Integrals

The trilogarithmic integral of an element g ∈ F −{0, 1} is deﬁned as Z g0 l (g) = l (g). 3 g 2

YK-VRS Integration with special functions 22 / 31 Trilogarithmic Integrals

The trilogarithmic integral of an element g ∈ F −{0, 1} is deﬁned as Z g0 l (g) = l (g). 3 g 2

A differential field extension E = F (θ1, . . . , θn) is called a transcendental TEL-extension if CE = CF and each θi is transcendental over Fi−1 and satisfies one of the following:

(i) θi is an exponential over Fi−1.

(ii) θi is a logarithm over Fi−1.

(iii) θi is an error function over Fi−1.

(iv) θi is a logarithmic integral over Fi−1.

(v) θi is a dilogarithmic integral over Fi−1.

(vi) θi is a trilogarithmic integral over Fi−1.

YK-VRS Integration with special functions 22 / 31 A T EL−expression is called (a)a special T EL−expression if for each i, 0 0 ri = ci log(1 − gi)gi/gi for some constant ci and for each l, 0 0 sl = dl(1 − hl) /(1 − hl) for some constant dl. (b)a T −expression if it is special and for all j, k, aj = bk = 0.

Integration with Trilogarithmic Integrals

We say that v ∈ F admits a T EL−expression over F if there are ﬁnite indexing sets I, J, K, L and elements −v2 ri, gi, sl, hl, uj, log(uj), vk, e k , w ∈ F and constants aj, bk for all i, j, k in I, J, K respectively such that

0 0 u0 X gi X hl X j X 0 −v2 0 v = ri + sl + aj + bkvke k + w , gi hl log(uj) i∈I l∈L j∈J k∈K

where for each i, l, there are integers ni, ml such that 0 Pni 0 0 Pml 0 ri = p=1 cilfil/fil and sl = q=1 dlq log(slq)tlq/tlq for some constants cip, dlq and elements fil, slq, tlq ∈ F

YK-VRS Integration with special functions 23 / 31 (b)a T −expression if it is special and for all j, k, aj = bk = 0.

Integration with Trilogarithmic Integrals

0 0 u0 X gi X hl X j X 0 −v2 0 v = ri + sl + aj + bkvke k + w , gi hl log(uj) i∈I l∈L j∈J k∈K

where for each i, l, there are integers ni, ml such that 0 Pni 0 0 Pml 0 ri = p=1 cilfil/fil and sl = q=1 dlq log(slq)tlq/tlq for some constants cip, dlq and elements fil, slq, tlq ∈ F

A T EL−expression is called (a)a special T EL−expression if for each i, 0 0 ri = ci log(1 − gi)gi/gi for some constant ci and for each l, 0 0 sl = dl(1 − hl) /(1 − hl) for some constant dl.

YK-VRS Integration with special functions 23 / 31 Integration with Trilogarithmic Integrals

0 0 u0 X gi X hl X j X 0 −v2 0 v = ri + sl + aj + bkvke k + w , gi hl log(uj) i∈I l∈L j∈J k∈K

where for each i, l, there are integers ni, ml such that 0 Pni 0 0 Pml 0 ri = p=1 cilfil/fil and sl = q=1 dlq log(slq)tlq/tlq for some constants cip, dlq and elements fil, slq, tlq ∈ F

A T EL−expression is called (a)a special T EL−expression if for each i, 0 0 ri = ci log(1 − gi)gi/gi for some constant ci and for each l, 0 0 sl = dl(1 − hl) /(1 − hl) for some constant dl. (b)a T −expression if it is special and for all j, k, aj = bk = 0. YK-VRS Integration with special functions 23 / 31 Integration with Trilogarithmic Integrals

We call a differential field extension E of F to be a dilogarithmic extension of F if they have same field of constants and there are elements y1, . . . , yn, z1, . . . , zm ∈ F such that E = F (log(y1),..., log(yn), `2(z1), . . . , `2(zm)).

YK-VRS Integration with special functions 24 / 31 Integration with Trilogarithmic Integrals

Lemma Let F (θ) ⊃ F be a transcendental trilogarithmic-elementary extension of F . Suppose there is an element v ∈ F such that v admits a special T EL−expression over a dilogarithmic extension E = F (θ)(log(y1),..., log(yn), `2(z1), . . . , `2(zm)) where each yi, zi ∈ F (θ). Then there exists a ﬁeld M = F (log(p1),..., log(pl), `2(q1), . . . , `2(qt), θ), where each pi, qi ∈ F, having the same ﬁeld of constants as F such that v admits a special T EL−expression over M.

YK-VRS Integration with special functions 25 / 31 J. Baddoura (2006) proved a similar identity for Bloch-Wigner Spence function D(g).

Integration with Trilogarithmic Integrals

Proposition Let F (θ) ⊃ F be a transcendental ﬁeld extension with CF (θ) = CF . Let f be any element in F (θ) and {αj; j = 1, . . . , t} be the set of all zeroes and poles of f and 1 − f in an algebraic closure of F . Also, for some integers aj, Qt aj Qt bj bj, let f = η j=1(θ − αj) and 1 − f = ξ j=1(θ − αj) then

t t X θ − αj 1 X 2 `2(f) = `2(η) − ajbk`2 − ajbk log (θ − αk) θ − αk 2 j,k=1 j,k=1 k6=j t t X X θ − αj − ak log(θ − αk) log ξ − ajbk log log(αj − αk). θ − αk k=1 j,k=1 k6=j

YK-VRS Integration with special functions 26 / 31 Integration with Trilogarithmic Integrals

J. Baddoura (2006) proved a similar identity for Bloch-Wigner Spence function D(g).

YK-VRS Integration with special functions 26 / 31 Integration with Trilogarithmic Integrals

Theorem Let E be a transcendental T EL−extension of F. Then an element u in E have u0 in F if and only if there are ﬁnite −v2 indexing sets I, J, K, L and w, gi, hl, ri, sl, uj, log(uj), vk, e k in F such that 0 0 u0 0 X gi X hl X j X 0 −v2 0 u = ri + sl + aj + bkvke k + w , gi hl log(uj) i∈I l∈L j∈J k∈K

0 0 0 0 0 gi X hl 0 X gi X hp ri = −citi − ril , sl = − ril − slp , gi hl gi hp l∈L i∈I p∈L

0 0 0 0 0 (1 − gi) X hl 0 gi X hp ti = − cil , ril = −cicil − eilp 1 − gi hl gi hp l∈L p∈L

YK-VRS Integration with special functions 27 / 31 Integration with Trilogarithmic Integrals

Theorem (Cont.) 0 0 0 X gi X hq and slp = − eilp − flpq , gi hq i∈I q∈L

where each ci is a non-zero constant, each cil, eilp, flpq are some constants and each ti, ril and slp are elements in some dilogarithmic extension of F with eilp = eipl and slp = spl for every l and p.

YK-VRS Integration with special functions 28 / 31 References

J. Baddoura, Integration in finite terms with elementary functions and dilogarithms, J. Symbolic Comput., Vol 41, No. 8 (2006), pp.909–942. J. Baddoura, A note on symbolic integration with polylogarithms, Mediterr. J. Math, Vol. 8, no. 2 (2011), pp. 229–241. G. Cherry Integration in finite terms with special functions: the logarithmic integral, SIAM J. Comput. Vol. 15, no. 1 (1986), pp. 1–21. G. Cherry, Integration in finite terms with special functions: the error function, J. Symbolic Comput. Vol 1, no. 3 (1985), pp. 283–302. E.R. Kolchin, Algebraic Groups and Algebraic Dependence, Amer. J. of Math, Vol. 90, No.4. (1968), pp.1151-1164. J. Moses, The integration of a Class of Special Functions with the Risch Algorithm, ACM SIGSAM Bulletin, No.13 (1969), pp.14-27. YK-VRS Integration with special functions 29 / 31 References

L. Rubel, M. Singer, Autonomous Functions, Jour. Diff eqns, Vol. 75, (1988), pp. 354-370. M. Rosenlicht, Liouville’s theorem on functions with elementary integrals, Pacific J. Math, Vol. 24, (1968), pp. 153–161. M. Rosenlicht, M. Singer, On Elementary, Generalized Elementary, and Liouvillian Extension Fields, Contributions to Algebra, (H. Bass et.al., ed.), Academic Press (1977), pp. 329-342. M. Singer, B. Saunders, B. Caviness, An extension of Liouville’s theorem on integration in finite terms, SIAM J. Comput., Vol. 14, no. 4 (1985), 966–990. Y. Kaur, V. Srinivasan, Integration in Finite Terms with Dilogarithmic Integrals, Logarithmic Integrals and Error Functions, J. Symb. Comput. (2018), https://doi.org/10.1016/j.jsc.2018.08.004

YK-VRS Integration with special functions 30 / 31 Thank You!

YK-VRS Integration with special functions 31 / 31