Generalization of Risch's Algorithm to Special Functions Clemens G. Raab (RISC) LHCPhenonet School Hagenberg, July 9, 2012 Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions Overview 1 Introduction to symbolic integration 2 Relevant classes of functions and Risch's algorithm 3 Basics of differential fields 4 A generalization of Risch's algorithm 1 Introduction 2 Inside the algorithm 5 Application to definite integrals depending on parameters Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions Introduction to symbolic integration Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions Different approaches and structures Differential algebra: differential fields Holonomic systems: Ore algebras Rule-based: expressions, tables of transformation rules ... Symbolic integration Computer algebra 1 Model the functions by algebraic structures 2 Computations in the algebraic framework 3 Interpret result in terms of functions Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions Holonomic systems: Ore algebras Rule-based: expressions, tables of transformation rules ... Symbolic integration Computer algebra 1 Model the functions by algebraic structures 2 Computations in the algebraic framework 3 Interpret result in terms of functions Different approaches and structures Differential algebra: differential fields Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions Rule-based: expressions, tables of transformation rules ... Symbolic integration Computer algebra 1 Model the functions by algebraic structures 2 Computations in the algebraic framework 3 Interpret result in terms of functions Different approaches and structures Differential algebra: differential fields Holonomic systems: Ore algebras Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions Symbolic integration Computer algebra 1 Model the functions by algebraic structures 2 Computations in the algebraic framework 3 Interpret result in terms of functions Different approaches and structures Differential algebra: differential fields Holonomic systems: Ore algebras Rule-based: expressions, tables of transformation rules ... Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions Examples Z Li (x) − xLi (x) x ln(1 − x)2 3 2 dx = (Li (x) − Li (x)) + (1 − x)2 1 − x 3 2 2 Z 0 2 1 0 2 0 2 2 Ai (x) dx = 3 xAi (x) + 2Ai(x)Ai (x) − x Ai(x) Z 1 π Y (x) dx = ln n xJn(x)Yn(x) 2 Jn(x) Indefinite integration Antiderivatives Z f (x) dx = g(x) Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions Indefinite integration Antiderivatives Z f (x) dx = g(x) Examples Z Li (x) − xLi (x) x ln(1 − x)2 3 2 dx = (Li (x) − Li (x)) + (1 − x)2 1 − x 3 2 2 Z 0 2 1 0 2 0 2 2 Ai (x) dx = 3 xAi (x) + 2Ai(x)Ai (x) − x Ai(x) Z 1 π Y (x) dx = ln n xJn(x)Yn(x) 2 Jn(x) Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions Examples Z 1 zx x dx = Li2(z) 0 e − z Z 1 −sx Γ(a) e γ(a; x) dx = a 0 s(s + 1) n Z 1 1 i X 1 e−2nπix ln(sin( π x)) dx = − + 2 4n nπ 2k − 1 0 k=1 Definite integration Integrals depending on parameters Z b f (x; y) dx = g(y) a Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions Definite integration Integrals depending on parameters Z b f (x; y) dx = g(y) a Examples Z 1 zx x dx = Li2(z) 0 e − z Z 1 −sx Γ(a) e γ(a; x) dx = a 0 s(s + 1) n Z 1 1 i X 1 e−2nπix ln(sin( π x)) dx = − + 2 4n nπ 2k − 1 0 k=1 Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions After integrating from 0 to 1 we obtain Z 1 Z 1 z −x 1 z f (z; x)dx − f (z + 1; x)dx = x e x=0 0 0 In other words, we proved zΓ(z) − Γ(z + 1) = 0 Example: Gamma function Z 1 Γ(z) := xz−1e−x dx for z > 0 0 | {z } =:f (z;x) We compute d zf (z; x) − f (z + 1; x) = xz e−x dx Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions In other words, we proved zΓ(z) − Γ(z + 1) = 0 Example: Gamma function Z 1 Γ(z) := xz−1e−x dx for z > 0 0 | {z } =:f (z;x) We compute d zf (z; x) − f (z + 1; x) = xz e−x dx After integrating from 0 to 1 we obtain Z 1 Z 1 z −x 1 z f (z; x)dx − f (z + 1; x)dx = x e x=0 0 0 Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions Example: Gamma function Z 1 Γ(z) := xz−1e−x dx for z > 0 0 | {z } =:f (z;x) We compute d zf (z; x) − f (z + 1; x) = xz e−x dx After integrating from 0 to 1 we obtain Z 1 Z 1 z −x 1 z f (z; x)dx − f (z + 1; x)dx = x e x=0 0 0 In other words, we proved zΓ(z) − Γ(z + 1) = 0 Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions yields an ODE for Z b I (y) := f (x; y)dx a d c0(n)f (x; n) + ··· + cm(n)f (x; n + m) = dx g(x; n) yields a recurrence for Z b I (n) := f (x; n)dx a Linear Relations Integrals depending on one parameter @mf d c0(y)f (x; y) + ··· + cm(y) @y m (x; y) = dx g(x; y) Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions d c0(n)f (x; n) + ··· + cm(n)f (x; n + m) = dx g(x; n) yields a recurrence for Z b I (n) := f (x; n)dx a Linear Relations Integrals depending on one parameter @mf d c0(y)f (x; y) + ··· + cm(y) @y m (x; y) = dx g(x; y) yields an ODE for Z b I (y) := f (x; y)dx a Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions yields a recurrence for Z b I (n) := f (x; n)dx a Linear Relations Integrals depending on one parameter @mf d c0(y)f (x; y) + ··· + cm(y) @y m (x; y) = dx g(x; y) yields an ODE for Z b I (y) := f (x; y)dx a d c0(n)f (x; n) + ··· + cm(n)f (x; n + m) = dx g(x; n) Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions Linear Relations Integrals depending on one parameter @mf d c0(y)f (x; y) + ··· + cm(y) @y m (x; y) = dx g(x; y) yields an ODE for Z b I (y) := f (x; y)dx a d c0(n)f (x; n) + ··· + cm(n)f (x; n + m) = dx g(x; n) yields a recurrence for Z b I (n) := f (x; n)dx a Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions g(x) is a certificate for the relation It is easy to verify 0 c0f0(x) + ··· + cmfm(x) = g (x) and r = g(b) − g(a) 0 ;:::; fm(x) and c0;:::; cm const. w.r.t. x c0 0 + ··· + cmfm(x) Transfer this to a relation of corresponding integrals b b R R c0 f0(x)dx + ··· + cm fm(x)dx = g(b) − g(a) a a Certificate b b R R c0 f0(x)dx + ··· + cm fm(x)dx = r a a Parametric integration Compute linear relation of integrals Given f (x) , find g(x) s.t. f (x) = g 0(x) Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions g(x) is a certificate for the relation It is easy to verify 0 c0f0(x) + ··· + cmfm(x) = g (x) and r = g(b) − g(a) and c0;:::; cm const. w.r.t. x c0 0 + ··· + cmfm(x) Transfer this to a relation of corresponding integrals b b R R c0 f0(x)dx + ··· + cm fm(x)dx = g(b) − g(a) a a Certificate b b R R c0 f0(x)dx + ··· + cm fm(x)dx = r a a Parametric integration Compute linear relation of integrals Given f 0(x);:::; fm(x), find g(x) s.t. f (x) = g 0(x) Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions g(x) is a certificate for the relation It is easy to verify 0 c0f0(x) + ··· + cmfm(x) = g (x) and r = g(b) − g(a) c0 0 + ··· + cmfm(x) Transfer this to a relation of corresponding integrals b b R R c0 f0(x)dx + ··· + cm fm(x)dx = g(b) − g(a) a a Certificate b b R R c0 f0(x)dx + ··· + cm fm(x)dx = r a a Parametric integration Compute linear relation of integrals Given f 0(x);:::; fm(x), find g(x) and c0;:::; cm const. w.r.t. x s.t. f (x) = g 0(x) Clemens G. Raab (RISC) Generalization of Risch's Algorithm to Special Functions g(x) is a certificate for the relation It is easy to verify 0 c0f0(x) + ··· + cmfm(x) = g (x) and r = g(b) − g(a) Transfer this to a relation of corresponding integrals b b R R c0 f0(x)dx + ··· + cm fm(x)dx = g(b) − g(a) a a Certificate b b R R c0 f0(x)dx + ··· + cm fm(x)dx = r a a Parametric integration Compute linear relation of integrals Given f 0(x);:::; fm(x), find g(x) and c0;:::; cm const.