FeynCalc manual (development version)

Integrate2

Integrate2 is like Integrate, but Integrate2[a_Plus, b__] := Map[Integrate2[#, b]&, a] ( more linear algebra and partial fraction decomposition is done)

Integrate2[f[x] DeltaFunction[x], x] -> f[0]

Integrate2[f[x] DeltaFunction[x0-x], x] -> f[x0]

Integrate2[f[x] DeltaFunction[a + b x], x] -> Integrate[f[x] (1/Abs[b]) DeltaFunction[a/b + x], x], where Abs[b] -> b, if b is a symbol, and if b = -c, then Abs[-c] -> c, i.e., the variable contained in b is supposed to be positive.

π2\pi ^2 is replaced by 6 Zeta2.

Integrate2[1/(1-y),{y,x,1}] is interpreted as distribution, i.e. as Integrate2[-1/(1-y)],{y, 0, x}] -> Log[1-y].

Integrate2[1/(1-x),{x,0,1}] -> 0

Since Integrate2 does do a reordering and partial fraction decomposition before calling the integral table of Integrate3, it will in general be slower compared to Integrate3 for sums of integrals. I.e., if the integrand has already an expanded form and if partial fraction decomposition is not necessary it is more effective to use Integrate3.

See also

Overview, DeltaFunction, Integrate3, Integrate5, SumS, SumT.

Examples

Integrate2[Log[1 + x] Log[x]/(1 - x), {x, 0, 1}] // Timing

{0.057955,ζ(3)32ζ(2)log(2)}\left\{0.057955,\zeta (3)-\frac{3}{2} \zeta (2) \log (2)\right\}

Since Integrate2 uses table-look-up methods it is much faster than Mathematica’s Integrate.

Integrate2[PolyLog[2, x^2], {x, 0, 1}]

ζ(2)4+4log(2)\zeta (2)-4+4 \log (2)

Integrate2[PolyLog[3, -x], {x, 0, 1}]

ζ(2)23ζ(3)4+12log(2)\frac{\zeta (2)}{2}-\frac{3 \zeta (3)}{4}+1-2 \log (2)

Integrate2[PolyLog[3, 1/(1 + x)], {x, 0, 1}]

ζ(2)(log(2))+3ζ(3)4+log3(2)3log2(2)+2log(2)\zeta (2) (-\log (2))+\frac{3 \zeta (3)}{4}+\frac{\log ^3(2)}{3}-\log ^2(2)+2 \log (2)

Integrate2[DeltaFunction[1 - x] f[x], {x, 0, 1}]

f(1)f(1)

Integrate2 does integration in a Hadamard sense, i.e., 01f(x)dx\int _0^1 \, f(x) \, d x means actually expanding the result of δ1δf(x)dx\int _{\delta }^{1-\delta} \, f(x) \, dx up to O(δ)\mathcal{O}(\delta ) and neglecting all δ\delta-dependent terms. E.g. δ1δ11xdx=log(1x)δ1δ=log(δ)+log(1)0\int_{\delta }^{1-\delta} \frac{1}{1-x} \, d x = - \log (1-x) \biggl |_{\delta }^{1-\delta } = -\log (\delta )+log (1) \Rightarrow 0

Integrate2[1/(1 - x), {x, 0, 1}]

00

In the physics literature sometimes the “+” notation is used. In FeynCalc the (frac11x)+\left(frac{1}{1-x} \right)_{+} is represented by PlusDistribution}[1/(1-x)] or just 1/(1-x)

Integrate2[PlusDistribution[1/(1 - x)], {x, 0, 1}]

00

Integrate2[PolyLog[2, 1 - x]/(1 - x)^2, {x, 0, 1}]

2ζ(2)2-\zeta (2)

Integrate2[(Log[x] Log[1 + x])/(1 + x), {x, 0, 1}]

ζ(3)8-\frac{\zeta (3)}{8}

Integrate2[Log[x]^2/(1 - x), {x, 0, 1}]

2ζ(3)2 \zeta (3)

Integrate2[PolyLog[2, -x]/(1 + x), {x, 0, 1}]

ζ(3)412ζ(2)log(2)\frac{\zeta (3)}{4}-\frac{1}{2} \zeta (2) \log (2)

Integrate2[Log[x] PolyLog[2, x], {x, 0, 1}]

32ζ(2)3-2 \zeta (2)

Integrate2[x PolyLog[3, x], {x, 0, 1}]

ζ(2)4+ζ(3)2+316-\frac{\zeta (2)}{4}+\frac{\zeta (3)}{2}+\frac{3}{16}

Integrate2[(Log[x]^2 Log[1 - x])/(1 + x), {x, 0, 1}]

ζ(4)+ζ(2)log2(2)4  Li4(12)log4(2)6\zeta (4)+\zeta (2) \log ^2(2)-4 \;\text{Li}_4\left(\frac{1}{2}\right)-\frac{\log ^4(2)}{6}

Integrate2[PolyLog[2, ((x (1 - z) + z) (1 - x + x z))/z]/(1 - x + x z), {x, 0, 1}]

2iπ  Li2(z)1z4  Li3(1z2)1z+4  Li3(1z)1z+2  Li3(z)1z+4  Li3(1z+1)1z4  Li3(1zz+1)1z4  Li3(z+12)1z2  Li2(1z)log(z)1z2  Li2(z)log(z)1z+4  Li2(z)log(1z)1z2S12(1z)1z+iπζ(2)1zζ(2)log(z)1z+2ζ(2)log(1z)1z+6ζ(2)log(z+1)1z4ζ(2)log(2)1z+2ζ(3)1z+log3(z)6(1z)+4log3(2)3(1z)log(1z)log2(z)1zlog(z+1)log2(z)1ziπlog2(z)2(1z)2log(1z)log2(z+1)1z2log2(2)log(1z)1z2log2(2)log(z+1)1z+4log(1z)log(z+1)log(z)1z+2iπlog(z+1)log(z)1z+4log(2)log(1z)log(z+1)1z\frac{2 i \pi \;\text{Li}_2(-z)}{1-z}-\frac{4 \;\text{Li}_3\left(\frac{1-z}{2}\right)}{1-z}+\frac{4 \;\text{Li}_3(1-z)}{1-z}+\frac{2 \;\text{Li}_3(-z)}{1-z}+\frac{4 \;\text{Li}_3\left(\frac{1}{z+1}\right)}{1-z}-\frac{4 \;\text{Li}_3\left(\frac{1-z}{z+1}\right)}{1-z}-\frac{4 \;\text{Li}_3\left(\frac{z+1}{2}\right)}{1-z}-\frac{2 \;\text{Li}_2(1-z) \log (z)}{1-z}-\frac{2 \;\text{Li}_2(-z) \log (z)}{1-z}+\frac{4 \;\text{Li}_2(-z) \log (1-z)}{1-z}-\frac{2 S_{12}(1-z)}{1-z}+\frac{i \pi \zeta (2)}{1-z}-\frac{\zeta (2) \log (z)}{1-z}+\frac{2 \zeta (2) \log (1-z)}{1-z}+\frac{6 \zeta (2) \log (z+1)}{1-z}-\frac{4 \zeta (2) \log (2)}{1-z}+\frac{2 \zeta (3)}{1-z}+\frac{\log ^3(z)}{6 (1-z)}+\frac{4 \log ^3(2)}{3 (1-z)}-\frac{\log (1-z) \log ^2(z)}{1-z}-\frac{\log (z+1) \log ^2(z)}{1-z}-\frac{i \pi \log ^2(z)}{2 (1-z)}-\frac{2 \log (1-z) \log ^2(z+1)}{1-z}-\frac{2 \log ^2(2) \log (1-z)}{1-z}-\frac{2 \log ^2(2) \log (z+1)}{1-z}+\frac{4 \log (1-z) \log (z+1) \log (z)}{1-z}+\frac{2 i \pi \log (z+1) \log (z)}{1-z}+\frac{4 \log (2) \log (1-z) \log (z+1)}{1-z}

Apart[Integrate2[x^(OPEm - 1) PolyLog[3, 1 - x], {x, 0, 1}], OPEm]

ζ(2)m2ζ(2)m1+ζ(2)+ζ(2)(S1(m2))+S12(m)+ζ(3)m-\frac{\zeta (2)}{m^2}-\frac{\zeta (2)}{m-1}+\frac{\zeta (2)+\zeta (2) \left(-S_1(m-2)\right)+S_{12}(m)+\zeta (3)}{m}

Integrate2[x^(OPEm - 1) Log[1 - x] Log[x] Log[1 + x]/(1 + x), {x, 0, 1}] // Simplify 
 
% /. OPEm -> 2 
 
N[%]

124(1)m(48ζ(4)+30ζ(2)log2(2)+6ζ(2)S12(m1)+18ζ(2)S2(m1)24ζ(2)S11(m1)12S2(m1)(ζ(2)log(4)S1(m1)log2(2))36ζ(2)log(2)S1(m1)+12S1(m1)(ζ(2)log(8)2ζ(3))+39ζ(3)S1(m1)+24S211(m1)+24S121(m1)+24S112(m1)+24S121(m1)+24S112(m1)+24S211(m1)12log2(2)S2(m1)+24log(2)S3(m1)24log(2)S21(m1)24log(2)S12(m1)48  Li4(12)63ζ(3)log(2)2log4(2))\frac{1}{24} (-1)^m \left(48 \zeta (4)+30 \zeta (2) \log ^2(2)+6 \zeta (2) S_{-1}^2(m-1)+18 \zeta (2) S_2(m-1)-24 \zeta (2) S_{1-1}(m-1)-12 S_{-2}(m-1) \left(\zeta (2)-\log (4) S_{-1}(m-1)-\log ^2(2)\right)-36 \zeta (2) \log (2) S_1(m-1)+12 S_{-1}(m-1) (\zeta (2) \log (8)-2 \zeta (3))+39 \zeta (3) S_1(m-1)+24 S_{-2-1-1}(m-1)+24 S_{-1-2-1}(m-1)+24 S_{-1-1-2}(m-1)+24 S_{1-21}(m-1)+24 S_{1-12}(m-1)+24 S_{2-11}(m-1)-12 \log ^2(2) S_2(m-1)+24 \log (2) S_3(m-1)-24 \log (2) S_{-21}(m-1)-24 \log (2) S_{-12}(m-1)-48 \;\text{Li}_4\left(\frac{1}{2}\right)-63 \zeta (3) \log (2)-2 \log ^4(2)\right)

124(48ζ(2)+48ζ(4)+30ζ(2)log2(2)+12(ζ(2)log2(2)+log(4))36ζ(2)log(2)48  Li4(12)12(ζ(2)log(8)2ζ(3))+39ζ(3)63ζ(3)log(2)1442log4(2)12log2(2)+72log(2))\frac{1}{24} \left(48 \zeta (2)+48 \zeta (4)+30 \zeta (2) \log ^2(2)+12 \left(\zeta (2)-\log ^2(2)+\log (4)\right)-36 \zeta (2) \log (2)-48 \;\text{Li}_4\left(\frac{1}{2}\right)-12 (\zeta (2) \log (8)-2 \zeta (3))+39 \zeta (3)-63 \zeta (3) \log (2)-144-2 \log ^4(2)-12 \log ^2(2)+72 \log (2)\right)

0.05051380.0505138

Integrate2[x^(OPEm - 1) (PolyLog[3, (1 - x)/(1 + x)] - PolyLog[3, -((1 - x)/(1 + x))]), {x, 0, 1}]

3ζ(2)(1)mlog(2)2m3ζ(2)log(2)2m+ζ(2)(1)mS1(m)mζ(2)S1(m)2m+ζ(2)(1)mS1(m)2mζ(2)S1(m)m+(1)mS3(m)m+(1)mS2(m)S1(m)m+S1(m)S2(m)m+S3(m)m(1)mS21(m)mS12(m)m(1)mS12(m)mS21(m)m7(1)mζ(3)8m+21ζ(3)8m\frac{3 \zeta (2) (-1)^m \log (2)}{2 m}-\frac{3 \zeta (2) \log (2)}{2 m}+\frac{\zeta (2) (-1)^m S_{-1}(m)}{m}-\frac{\zeta (2) S_{-1}(m)}{2 m}+\frac{\zeta (2) (-1)^m S_1(m)}{2 m}-\frac{\zeta (2) S_1(m)}{m}+\frac{(-1)^m S_{-3}(m)}{m}+\frac{(-1)^m S_{-2}(m) S_1(m)}{m}+\frac{S_1(m) S_2(m)}{m}+\frac{S_3(m)}{m}-\frac{(-1)^m S_{-21}(m)}{m}-\frac{S_{-1-2}(m)}{m}-\frac{(-1)^m S_{-12}(m)}{m}-\frac{S_{21}(m)}{m}-\frac{7 (-1)^m \zeta (3)}{8 m}+\frac{21 \zeta (3)}{8 m}

DataType[OPEm, PositiveInteger] 
 
Integrate2[x^(OPEm - 1) DeltaFunction[1 - x], {x, 0, 1}]

True\text{True}

11

This is the polarized non-singlet spin splitting function whose first moment vanishes.

t = SplittingFunction[PQQNS] /. FCGV[z_] :> ToExpression[z]

8CF(CFCA2)((x2+1)(2ζ(2)4  Li2(x)+log2(x)4log(x+1)log(x))x+1+4(1x)+2(x+1)log(x))+CACF(4(x2+1)log2(x)1x+8ζ(2)(x+1)+(536916ζ(2))(11x)++δ(1x)(88ζ(2)324ζ(3)+173)+49(53187x)43(5x221x+5)log(x))+CFNf(8(x2+1)log(x)3(1x)+(16ζ(2)323)δ(1x)+88x9809(11x)+89)+CF2(16(x2+1)log(1x)log(x)1x+δ(1x)(24ζ(2)+48ζ(3)+3)40(1x)4(x+1)log2(x)8(2x+31x)log(x))-8 C_F \left(C_F-\frac{C_A}{2}\right) \left(\frac{\left(x^2+1\right) \left(-2 \zeta (2)-4 \;\text{Li}_2(-x)+\log ^2(x)-4 \log (x+1) \log (x)\right)}{x+1}+4 (1-x)+2 (x+1) \log (x)\right)+C_A C_F \left(\frac{4 \left(x^2+1\right) \log ^2(x)}{1-x}+8 \zeta (2) (x+1)+\left(\frac{536}{9}-16 \zeta (2)\right) \left(\frac{1}{1-x}\right)_++\delta (1-x) \left(\frac{88 \zeta (2)}{3}-24 \zeta (3)+\frac{17}{3}\right)+\frac{4}{9} (53-187 x)-\frac{4}{3} \left(5 x-\frac{22}{1-x}+5\right) \log (x)\right)+C_F N_f \left(-\frac{8 \left(x^2+1\right) \log (x)}{3 (1-x)}+\left(-\frac{16 \zeta (2)}{3}-\frac{2}{3}\right) \delta (1-x)+\frac{88 x}{9}-\frac{80}{9} \left(\frac{1}{1-x}\right)_+-\frac{8}{9}\right)+C_F^2 \left(-\frac{16 \left(x^2+1\right) \log (1-x) \log (x)}{1-x}+\delta (1-x) (-24 \zeta (2)+48 \zeta (3)+3)-40 (1-x)-4 (x+1) \log ^2(x)-8 \left(2 x+\frac{3}{1-x}\right) \log (x)\right)

t // Expand

8ζ(2)CACF16x2CACF  Li2(x)x+116CACF  Li2(x)x+18ζ(2)x2CACFx+1+4x2CACFlog2(x)1x+4x2CACFlog2(x)x+116x2CACFlog(x)log(x+1)x+1+883ζ(2)CACFδ(1x)+173CACFδ(1x)+8ζ(2)xCACF8ζ(2)CACFx+116ζ(2)(11x)+CACF24ζ(3)CACFδ(1x)8929xCACF+5369(11x)+CACF+4CACFlog2(x)1x+4CACFlog2(x)x+1+43CACFlog(x)+43xCACFlog(x)+88CACFlog(x)3(1x)16CACFlog(x)log(x+1)x+1+356CACF98x2CFNflog(x)3(1x)163ζ(2)CFNfδ(1x)23CFNfδ(1x)+889xCFNf809(11x)+CFNf8CFNflog(x)3(1x)8CFNf9+32x2CF2  Li2(x)x+1+32CF2  Li2(x)x+1+16ζ(2)x2CF2x+18x2CF2log2(x)x+116x2CF2log(1x)log(x)1x+32x2CF2log(x)log(x+1)x+124ζ(2)CF2δ(1x)+3CF2δ(1x)+16ζ(2)CF2x+1+48ζ(3)CF2δ(1x)+72xCF24xCF2log2(x)8CF2log2(x)x+14CF2log2(x)32xCF2log(x)16CF2log(1x)log(x)1x24CF2log(x)1x16CF2log(x)+32CF2log(x)log(x+1)x+172CF28 \zeta (2) C_A C_F-\frac{16 x^2 C_A C_F \;\text{Li}_2(-x)}{x+1}-\frac{16 C_A C_F \;\text{Li}_2(-x)}{x+1}-\frac{8 \zeta (2) x^2 C_A C_F}{x+1}+\frac{4 x^2 C_A C_F \log ^2(x)}{1-x}+\frac{4 x^2 C_A C_F \log ^2(x)}{x+1}-\frac{16 x^2 C_A C_F \log (x) \log (x+1)}{x+1}+\frac{88}{3} \zeta (2) C_A C_F \delta (1-x)+\frac{17}{3} C_A C_F \delta (1-x)+8 \zeta (2) x C_A C_F-\frac{8 \zeta (2) C_A C_F}{x+1}-16 \zeta (2) \left(\frac{1}{1-x}\right)_+ C_A C_F-24 \zeta (3) C_A C_F \delta (1-x)-\frac{892}{9} x C_A C_F+\frac{536}{9} \left(\frac{1}{1-x}\right)_+ C_A C_F+\frac{4 C_A C_F \log ^2(x)}{1-x}+\frac{4 C_A C_F \log ^2(x)}{x+1}+\frac{4}{3} C_A C_F \log (x)+\frac{4}{3} x C_A C_F \log (x)+\frac{88 C_A C_F \log (x)}{3 (1-x)}-\frac{16 C_A C_F \log (x) \log (x+1)}{x+1}+\frac{356 C_A C_F}{9}-\frac{8 x^2 C_F N_f \log (x)}{3 (1-x)}-\frac{16}{3} \zeta (2) C_F N_f \delta (1-x)-\frac{2}{3} C_F N_f \delta (1-x)+\frac{88}{9} x C_F N_f-\frac{80}{9} \left(\frac{1}{1-x}\right)_+ C_F N_f-\frac{8 C_F N_f \log (x)}{3 (1-x)}-\frac{8 C_F N_f}{9}+\frac{32 x^2 C_F^2 \;\text{Li}_2(-x)}{x+1}+\frac{32 C_F^2 \;\text{Li}_2(-x)}{x+1}+\frac{16 \zeta (2) x^2 C_F^2}{x+1}-\frac{8 x^2 C_F^2 \log ^2(x)}{x+1}-\frac{16 x^2 C_F^2 \log (1-x) \log (x)}{1-x}+\frac{32 x^2 C_F^2 \log (x) \log (x+1)}{x+1}-24 \zeta (2) C_F^2 \delta (1-x)+3 C_F^2 \delta (1-x)+\frac{16 \zeta (2) C_F^2}{x+1}+48 \zeta (3) C_F^2 \delta (1-x)+72 x C_F^2-4 x C_F^2 \log ^2(x)-\frac{8 C_F^2 \log ^2(x)}{x+1}-4 C_F^2 \log ^2(x)-32 x C_F^2 \log (x)-\frac{16 C_F^2 \log (1-x) \log (x)}{1-x}-\frac{24 C_F^2 \log (x)}{1-x}-16 C_F^2 \log (x)+\frac{32 C_F^2 \log (x) \log (x+1)}{x+1}-72 C_F^2

Integrate2[t, {x, 0, 1}] // Timing

{0.040008,0}\{0.040008,0\}

Expanding t with respect to x yields a form already suitable for Integrate3 and therefore the following is faster:

Integrate3[Expand[t, x], {x, 0, 1}] // Expand // Timing

{0.018181,0}\{0.018181,0\}

Clear[t]; 
 
Integrate2[DeltaFunction[1 - x] f[x], {x, 0, 1}]

f(1)f(1)

Integrate2[x^5 Log[1 + x]^2, {x, 0, 1}] 
 
N[%]

46log(2)45695910800\frac{46 \log (2)}{45}-\frac{6959}{10800}

0.06419860.0641986

NIntegrate[x^5 Log[1 + x]^2, {x, 0, 1}]

0.06419860.0641986

Integrate2[x^(OPEm - 1) Log[1 + x]^2, {x, 0, 1}]

2(1)mS12(m)m+(1)mS1(m12)S1(m)mS1(m12)S1(m)m+(1)mS1(m2)S1(m)m+S1(m2)S1(m)m+(1)mS2(m12)2mS2(m12)2m+(1)mS2(m2)2m+S2(m2)2m2(1)mS2(m)m2(1)mS11(m)m+4(1)mlog(2)S1(m)m(1)mlog(2)S1(m12)m+log(2)S1(m12)m(1)mlog(2)S1(m2)mlog(2)S1(m2)m(1)mlog2(2)m+log2(2)m-\frac{2 (-1)^m S_1^2(m)}{m}+\frac{(-1)^m S_1\left(\frac{m-1}{2}\right) S_1(m)}{m}-\frac{S_1\left(\frac{m-1}{2}\right) S_1(m)}{m}+\frac{(-1)^m S_1\left(\frac{m}{2}\right) S_1(m)}{m}+\frac{S_1\left(\frac{m}{2}\right) S_1(m)}{m}+\frac{(-1)^m S_2\left(\frac{m-1}{2}\right)}{2 m}-\frac{S_2\left(\frac{m-1}{2}\right)}{2 m}+\frac{(-1)^m S_2\left(\frac{m}{2}\right)}{2 m}+\frac{S_2\left(\frac{m}{2}\right)}{2 m}-\frac{2 (-1)^m S_2(m)}{m}-\frac{2 (-1)^m S_{-11}(m)}{m}+\frac{4 (-1)^m \log (2) S_1(m)}{m}-\frac{(-1)^m \log (2) S_1\left(\frac{m-1}{2}\right)}{m}+\frac{\log (2) S_1\left(\frac{m-1}{2}\right)}{m}-\frac{(-1)^m \log (2) S_1\left(\frac{m}{2}\right)}{m}-\frac{\log (2) S_1\left(\frac{m}{2}\right)}{m}-\frac{(-1)^m \log ^2(2)}{m}+\frac{\log ^2(2)}{m}