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Monday, February 28, 2022

Integral of the day XXVII

An integral involving the Gamma function

Integral involving the Gamma functon Γ()


Today we show the proof of this integral posted by @Ali39342137 0(1+x4x2)nxk1dx=Γ(n2k4)Γ(k2)2k2Γ(n2+k4)n2n+kn>k2 This is a generalization for this other integral posted by @integralsbot with n=s and k=1: 0(1+x4x2)sdx=s2s+1π2Γ(s214)Γ(s2+14) We previously posted another similar generalization here

Proof:

I=0(1+x4x2)nxk1dx=120(1+w2w)nwk21dw(wx2)=12k2+10tnk21(1t2)k21(t2+1)(t1+w2w)=12k2+1[0tnk2+1(1t2)k21dt+0tnk21(1t2)k21dt]=12k2+2[0sn2k4(1s)k21ds+0sn2k41(1s)k21ds](st2)=12k2+2[B(n2k4+1,k2)+B(n2k4,k2)]=12k2+2[Γ(n2k4+1)Γ(k2)Γ(n2+k4+1)+Γ(n2k4)Γ(k2)Γ(n2+k4)]=12k2+2[(n2k4)Γ(n2k4)Γ(k2)(n2+k4)Γ(n2+k4)+Γ(n2k4)Γ(k2)Γ(n2+k4)]=Γ(n2k4)Γ(k2)2k2+2Γ(n2+k4)[(n2k4)(n2+k4)+1]=Γ(n2k4)Γ(k2)2k2Γ(n2+k4)n2n+k 0(1+x4x2)nxk1dx=Γ(n2k4)Γ(k2)2k2Γ(n2+k4)n2n+kn>k2

Thursday, February 17, 2022

Golden ratio II

An integral involving the Dirichlet eta function

Nice hypergeometric series involving the Fibonacci numbers F2k+3


Today we show the proof of this nice series proposed by @diegorattaggi n=1(n+k1k)Fn2n=2kF2k+3 For the proof we will use some properties of the rising factorial (Pochhammer polynomial), the Gaussian hypergeometric function and the Binet's formula for Fibonacci numbers.

Proof:

S=n=1(n+k1k)Fn2n=n=1(n+k1)!(n1)!k!Fn2n=n=1Γ(n+k)(n1)!k!Fn2n Recall that the pochhammer polynomial (rising factorial) can be expressed as the quotient of two gamma functions: Γ(n+x)Γ(x)=(x)n where (x)n=x(x+1)(x+2)(x+n1) Hence S=n=1Γ(n+k)(n1)!k!Fn2n=Γ(k)k!n=1(k)n(n1)!Fn2n=1kn=1(k)n(n1)!Fn2n If we let the change of variable n=j+1 S=12kj=0(k)j+1j!Fj+12j From the recursion formula for the degree of the Pochhammer polynomial: (x)n+1=x(x+1)n we have S=12kj=0(k)j+1j!Fj+12j=12j=0(k+1)jj!Fj+12j From the Binet's formula Fj+1=ϕj+1ψj+15 S=ϕ25j=0(k+1)jj!ϕj2jψ25j=0(k+1)jj!ψj2j Multiplying and dividing by (1)j we have an hypergeometric representation: S=ϕ25j=0(k+1)jj!ϕj2jψ25j=0(k+1)jj!ψj2j=ϕ25j=0(k+1)j(1)j(1)j(ϕ2)jj!ψ25j=0(k+1)j(1)j(1)j(ψ2)jj!=ϕ25F(1,k+1;1;ϕ2)ψ25F(1,k+1;1;ψ2) This is a special case of the Gaussian hypergeometric fucntion (generalized Binomial theorem): F(a,b;a;z)=(1z)b If we put a=1,b=k+1 S=ϕ25F(1,k+1;1;ϕ2)ψ25F(1,k+1;1;ψ2)=ϕ25(1ϕ2)k1ψ25(1ψ2)k1 Note ϕ25(1ϕ2)k1=ϕ25(22ϕ)k+1=ϕ25(3+5)k+1=ϕ25(2+2ϕ)k+1=ϕ2k+125(1+ϕ)k+1=ϕ2k+125(ϕ2)k+1(1+ϕ=ϕ2)=2kϕ2k+35 ψ25(1ψ2)k1=ψ25(11ϕ2)k1(ψ=1ϕ)=ψ25(21+ϕ)k+1=2kψ5(1ϕ2)k+1(ϕ+1=ϕ2)=2kψ5(ψ2)k+1=2kψ2k+35 Hence S=2kϕ2k+352k5ψ2k+3=2k[ϕ2k+35ψ2k+35]=2kF2k+3 Therefore n=1(n+k1k)Fn2n=2kF2k+3

Sunday, February 13, 2022

Integral of the day XXVI

An integral involving the Dirichlet eta function

Integral involving the Dirichlet eta function η(v)


Today we show the proof of this integral proposed by @Ali39342137: 0(1x1sinhx)ln(x)xdx=γln(2)+12ln2(2)+ln(π)ln(2) For the proof we use the expansion in partial fractions of the hyperbolic cosecant function and the derivative of the Dirichlet eta function.

Proof:

Recall that the hyperbolic cosecant can be expanded as partial fractions: cschx=1sinhx=1x2xπ2+x2+2x4π2+x22x9π2+x2+=1x+2j=1(1)jxj2π2+x2 Hence, 0(1x1sinhx)ln(x)xdx=0(1x1x2j=1(1)jxj2π2+x2)ln(x)xdx=20j=1(1)jln(x)j2π2+x2dx=2j=10(1)jln(x)j2π2+x2dx=2π2j=1(1)jj20ln(x)1+(xjπ)2dx=2πj=1(1)jj0ln(jπw)1+w2dx=2πj=1(1)jj0ln(jπ)+ln(w)1+w2dw=2πj=1(1)jj[ln(jπ)011+w2dw+0ln(w)1+w2dw]=2πj=1(1)jj[π2ln(jπ)+0ln(w)1+w2dw]=2πj=1(1)jj[π2ln(jπ)+0ddt|t=0+wt1+w2dw]=2πj=1(1)jj[π2ln(jπ)+ddt|t=0+0wt1+w2dw] Recall the integral representation of the secant function: sec(x)=2π0y2xπy2+1dy|x|<π2 If we put x=tπ2 sec(tπ2)=2π0yty2+1dy Hence 0yty2+1dy=π2sec(tπ2) Therefore I=2πj=1(1)jj[π2ln(jπ)+ddt|t=0+0wt1+w2dw]=2πj=1(1)jj[π2ln(jπ)+ddt|t=0+π2sec(tπ2)]=2πj=1(1)jj[π2ln(jπ)+limt0+π24sec(tπ2)tan(tπ2)=0]=j=1(1)j+1ln(jπ)j=j=1(1)j+1[ln(j)+ln(π)]j=j=1(1)j+1ln(j)j+ln(π)j=1(1)j+1j=j=1(1)j+1ln(j)j+ln(π)ln(2) Recall the definition of the Dirichlet eta funcion: η(v)=j=1(1)j+1jv Taking the derivative η(v)=j=1(1)jln(j)jv Therefore limv1η(v)=j=1(1)jln(j)j Now recall the relationship: η(v)=(121V)ζ(v) Hence η(v)=2v[(2v2)ζ(v)+ln(4)ζ(v)] Taking the limit as v1 can be proven that: limv1η(v)=γln(2)12ln2(2) Therefore j=1(1)jln(j)j=γln(2)12ln2(2) Then we have I=j=1(1)j+1ln(j)j+ln(π)ln(2)=γln(2)+12ln2(2)+ln(π)ln(2) Therefore we can conclude 0(1x1sinhx)ln(x)xdx=γln(2)+12ln2(2)+ln(π)ln(2)

Saturday, February 12, 2022

Integral of the day XXV

An integral involving the reciprocal beta function

Integral related to the reciprocal beta function


Today we show the proof of this integral posted by @integralsbot π0x(sin(x2)cos(x2))sinxdx=2πln(2) The proof will rely on the integral representation of the reciprocal beta function.

Proof:

Recall the following integral representation of the reciprocal beta function (Proof in the Appendix): π0sinv1xsin(ax)dx=πsin(πa2)2v1vB(1+a+v2,1a+v2)v,aR,v>0 π0sinv1xcos(ax)dx=πcos(πa2)2v1vB(1+a+v2,1a+v2)v,aR,v>0 If we put v=12 π0sin(ax)sinxdx=πsin(πa2)2121vB(1+a2+14,1a2+14) π0cos(ax)sinxdx=πcos(πa2)2121vB(1+a2+14,1a2+14) Differentiating with respect to a: ddaπ0sin(ax)sinxdx=π0xcos(ax)sinxdx=2π2cos(πa2)+2πsin(πa2)ψ(34a2)2πsin(πa2)ψ(a2+34)B(a2+34,34a2) ddaπ0cos(ax)sinxdx=π0xsin(ax)sinxdx=2π2sin(πa2)2πcos(πa2)ψ(34a2)+2πcos(πa2)ψ(a2+34)B(a2+34,34a2) Taking the limit as a12 π0xcos(x2)sinxdx=πγ+π2+πψ(12)2 π0xsin(x2)sinxdx=πγπ2+πψ(12)2 From the formula ψ(12J)=ψ(12+J)=γln(4)+Jj=122j1J=0,1,2,... if we put J=0 we have ψ(12)+γ=ln(4) Therefore π0xcos(x2)sinxdx=π2πln(4)2 π0xsin(x2)sinxdx=π2πln(4)2 Hence π0x(sin(x2)cos(x2))sinxdx=π22+πln(4)2π22+πln(4)2=2πln(2) Therefore we can conclude π0x(sin(x2)cos(x2))sinxdx=2πln(2)

Appendix:

Here is the proof of the following integrals with a,bR: π0sinv1xsin(ax)dx=πsin(πa2)2v1vB(1+a+v2,1a+v2)v>0 π0sinv1xcos(ax)dx=πcos(πa2)2v1vB(1+a+v2,1a+v2)v>0

Proof:

Let a,bR I=π0sinv1xeiaxdx=π0eiax(eix+eix2i)v1dx=1(2i)v1π0eiaxj=0(v1j)(1)jeix(v12j)dx=1(2i)v1j=0(v1j)(1)jπ0eix(a+v12j)dx=1(2i)v1j=0(v1j)(1)ji(a+v12j)iπ(a+v12j)0ewdw=1(2i)v1j=0(v1j)(1)j(eiπ(a+v12j)1)i(a+v12j)=1(2i)v1j=0(v1j)(1)j(eiπ(a+v1)1)i(a+v12j)=(eiπ(a+v1)1)2v1ivj=0(v1j)(1)j(a+v12j)=(eiπ(a+v1)1)2v1ivj=0(vj)jj!(1)j(a+v12j)=(eiπ(a+v1)1)2v1ivj=0(1v)j(a+v12j)j! Note that 2jav+1=(a+v1)(3av2)j(av+12)j Hence I=(eiπ(a+v1)1)2v1ivj=0(1v)j(a+v12j)j!=(eiπ(a+v1)1)2v1iv1(a+v1)j=0(1v)j(av+12)j(3av2)j=(eiπ(a+v1)1)2v1iv1(a+v1)F(1v,av+12;3av2;1) From the Gauss's Hypergeometric Theorem: F(a,b;c;1)=Γ(c)Γ(cab)Γ(ca)Γ(cb)c>a+b Therefore, supposing v>0: I=(eiπ(a+v1)1)2v1iv1(a+v1)Γ(3av2)Γ(v)Γ(1a+v2)=(eiπ(a+v1)1)2viv1(1av)2(1av2)Γ(1av2)Γ(v)Γ(1a+v2)=(eiπ(a+v1)1)2vivΓ(1av2)Γ(v)Γ(1a+v2) Note (eiπ(a+v1)1)iv=2sin(πa2)cos(π(a+v)2) (eiπ(a+v1)1)iv=2cos(πa2)sin(π(a+v)2) and recall the Euler's reflection formula for cos(πt): Γ(12+t)Γ(12t)=πcos(πt) Therefore π0sinv1xsin(ax)dx=(π0sinv1xeiaxdx)=sin(πa2)cos(π(a+v)2)2v1Γ(1av2)Γ(v)Γ(1a+v2)=sin(πa2)2v1πΓ(1av2)Γ(v)Γ(1+a+v2)Γ(1a+v2)Γ(1a+v2)(Euler's reflection formula)=sin(πa2)2v1πΓ(v)Γ(1+a+v2)Γ(1a+v2)=πsin(πa2)2v1vB(1+a+v2,1a+v2) In a very similar way we can show : π0sinv1xcos(ax)dx=(π0sinv1xeiaxdx)=πcos(πa2)2v1vB(1+a+v2,1a+v2)

Monday, February 7, 2022

Generalized hypergeometric functions VIII

Integral related to the Gauss function

Nice integral related to the Gauss hypergeometric function F(a,b;c;z)


Today we show the proof of this nice integral without contour integration as proposed by @AlbahariRicardo. π20cos(aθ)cosb(θ)dθ=πΓ(b+1)2b+1Γ(2a+b2)Γ(2+a+b2)a,bC,(b)>1 The proof relies on the Gauss theorem for the function that have his name F(a,b;c;z), the proof of this theorem neither relies on contour integration.

Update: We updated this post to cover the case a,bC.

Proof:

Let a,bC I=π20cos(aθ)cosb(θ)dθ=12π20(eiaθ+eiaθ)(eiθ+eiθ2)bdθ=12π20eiaθ(eiθ+eiθ2)bdθ+12π20eiaθ(eiθ+eiθ2)bdθ=12π20eiaθ(eiθ+eiθ2)bdθ12π20eiaϑ(eiϑ+eiϑ2)bdϑϑθ=12π20eiaθ(eiθ+eiθ2)bdθ+120π2eiaϑ(eiϑ+eiϑ2)bdϑ=12π2π2eiaθ(eiθ+eiθ2)bdθ=12b+1π2π2eiaθj=0(bj)eiθ(b2j)dθ(Generalized binomial theorem)=12b+1j=0(bj)π2π2eiθ(a+b2j)dθ=12b+1j=0(bj)1i(a+b2j)iπ(a+b2j)2iπ(a+b2j)2ewdw(wiθ(a+b2j),a+b2jjN)=12b+1j=0(bj)eiπ(a+b2j)2eiπ(a+b2j)2i(a+b2j)=12b+1j=0(bj)eiπ(a+b)2eiπjeiπ(a+b)2eiπji(a+b2j)=12bj=0(bj)(1)j(a+b2j)eiπ(a+b)2eiπ(a+b)22i=sin(π(a+b)2)2bj=0(bj)(1)j(a+b2j)=sin(π(a+b)2)2bj=0(bj+1)jj!(1)j(a+b2j)=sin(π(a+b)2)2bj=0(b)j(a+b2j)j! Note that 2jab=(a+b)(2ab2)j(ab2)j Hence I=sin(π(a+b)2)2bj=0(b)j(a+b2j)j!=sin(π(a+b)2)2b(a+b)j=0(b)j(ab2)j(2ab2)j1j!=sin(π(a+b)2)2b(a+b)F(b,ab2;2ab2;1) where F(a,b;c;z) is the Gauss hypergeometric function which satisfies th Gauss theorem: F(a,b;c;1)=Γ(c)Γ(cab)Γ(ca)Γ(cb)(ca+b)>0 Applying the theorem to F(b,ab2;2ab2;1) and supposing (b)>1 F(b,ab2;2ab2;1)=Γ(2ab2)Γ(b+1)Γ(2a+b2)Γ(1) Hence I=sin(π(a+b)2)2b(a+b)Γ(2ab2)Γ(b+1)Γ(2a+b2)=12b(a+b)[πΓ(2a+b2)Γ(a+b2)]Γ(2ab2)Γ(b+1)Γ(2a+b2)(Euler reflection formula)=πΓ(b+1)2b(a+b)Γ(2a+b2)Γ(a+b2)=πΓ(b+1)2b+1Γ(2a+b2)Γ(2+a+b2) The cases b=a=2n, 2m=a>b=2n and 2m=a<b=2n can easily be obtained as limiting cases of π20cos(aθ)cosb(θ)dθ=πΓ(b+1)2b+1Γ(2a+b2)Γ(2+a+b2)(b)>1,a+b2jjN Hence, we can conclude π20cos(aθ)cosb(θ)dθ=πΓ(b+1)2b+1Γ(2a+b2)Γ(2+a+b2)a,bC,(b)>1

Wednesday, February 2, 2022

Integral of the day XXIV

Another integral for Lobachvesky

A miserious integral related to Lobachevsky


Today we show the proof of this strange integral posted by @integralsbot 0sin(tan(x))xdx=π2(11e) The trick to solve this integral is a variation to the Lobachevsky integral formula for odd functions and the application of the residue theorem.

Proof:

Recall the Lobachevsky integral formula for odd functions:

Theorem. Let f(x) an odd, complex-valued, π-periodic function such that f(x)tanx is absolutely integrable over a single period.

Therefore 0f(x)xdx=π20f(x)tan(x)dx Back to the integral:

Note that the function f(x)=sin(tan(x)) is an odd function that satisfies all the hypothesis of the theorem.

Hence I=0sin(tan(x))xdx=π20sin(tan(x))tan(x)dx from (*) =π20sin(w)w(1+w2)dw(wtanx)=0sin(w)wdw0wsin(w)1+w2dw=0sin(w)wdw12wsin(w)1+w2dw(the integrand is even)=0sin(w)wdwDirichlet integral(weiw1+w2dwJ)=π2(weiw1+w2dwJ) The first integral is the classic Dirichlet integral, the second is a little bit more difficult: Let R>1 and consider the following complex function f(z)=zeiz1+z2 This function has two poles at z=±i.

Now consider the following contour (half upper circle): C1=tt(R,R) C2=Reiθθ[0,π] Hence C=C1C2zeiz1+z2dz=C1f(z)dz+C2f(dz)dz=RRteit1+t2dt+π0iR2e2iθeiReiθ1+Reiθdθ We will show that the integral in the right hand side will vanish as R. For that end we will make use of the Jordan's Lemma:

Jordan's lemma. Consider a complex-valued, continuous function f, defined on a semicircular contour CR={Reiθθ[0,π]} of positive radius R lying in the upper half-plane, centered at the origin. If the function f is of the form f(z)=eiazg(z),zCR, with a positive parameter a, then Jordan's lemma states the following upper bound for the contour integral: |CRf(z)dz|πaMRwhereMR:=maxθ[0,π]|g(Reiθ)| with equality when g vanishes everywhere, in which case both sides are identically zero.

Applying the lemma to f(z): f(z)=zeiz1+z2=eizg(z) where g(z)=z1+z2 Hence |C2zeiz1+z2dz|πMR where MR=maxθ[0,π]|g(Reiθ)|=maxθ[0,π]|Reiθ1+R2ei2θ|=maxθ[0,π]R|1+R2ei2θ|=maxθ[0,π]R|1(R2ei2θ)|R|1R2|=RR21(R>1) Therefore |C2zeiz1+z2dz|πMRπRR21 Taking the limit as R C2zeiz1+z2dz=π0iR2e2iθeiReiθ1+Reiθdθ=0 Hence, since the only pole in the upper half circle is z=i we have by the residue theorem J=teit1+t2dt=Czeiz1+z2dz=2πiRes(f,i)=limzi2πizeiz(z+i)=πie Therefore I=π2(J)=π212(πie)=π2π2e=π2(11e) Hence, we can conclude 0sin(tan(x))xdx=π2(11e)

Integral of the day XXIII

An integral involving the Lobachvesky integral formula

Nice integral solved with contour integration


Today we show the proof of this fun integral posted by @integralsbot π4011+1tan2(x)dx=1+224π1 The proof will rely on contour integration round the unit complex circle.

Proof

I=π4011+1tan2(x)dx=101(1+1w2)(1+w2)dw(wtanx)=10(11w211w2)1(1+1w2)(1+w2)dw=1011w2w2(1+w2)dw=1011w2w2dwJ1011+w2dwK+101w21+w2dwL J=1011w2w2dwIBP=1w21w|10+21011w2dx=1+2arcsin(1)=1+π2 K=1011+w2dw=arctan(1)=π4 L=101w21+w2dw=12111w21+w2dw(the integrand is even)=π0sin2s1+cos2sds=π01cos2s1+cos2sds=π02(1+cos2s)1+cos2sds=2π011+cos2sdsπ0ds=4π013+cos(2s)dsπ(cos2ϑ=1+cos(2ϑ)2)=22π013+cosθdθMπ To solve M we can make the following change of variable cosθ=z+z12 dθ=dzzi with this change the integral is transformed in a contour integral round the unit complex circle M=2π013+cosθdθ=2i|z|=11z2+6z+1dz The only pole inside the unit circle is z=223. Therefore by the residue theorem: |z|=11z2+6z+1dz=2πiRes(1z2+6z+1,223)=limz2232πiz+3+22=πi2 Hence M=2i|z|=11z2+6z+1dz=2π2 Therefore L=22ππ I=Jk+L=1+π2π4+22ππ=1+224π1 Finally, we can conclude π4011+1tan2(x)dx=1+224π1

Series of the day

Series involving the digamma and the zeta functions The sum 1(n+1)pnq ...