Laplace transform related to an infinite product
We show the solution of this integral posted on Twitter by @diegorattaggi. The solution turned out to be the logarithm of the golden ratio. We also find a general formulation for this integral wich turned out to be a Laplace transform with a beutiful solution. ∫∞0e−x−e−2x−e−3x+e−4xx(1−e−5x)dx=ln(1+√52)
As a corollary we have the Laplace transform:
∫∞0e−sx[e−x−e−2x−e−3x+e−4xx(1−e−5x)]dx=ln[Γ(1+s5)Γ(4+s5)Γ(2+s5)Γ(3+s5)]
Moreover, the integral and the Laplace transform can be further generalized:
If a1+a2+...+an=b1+b2+...+bn=c with ai,bi∈N∖{0}
Hence ∫∞0e−a1x+e−a2x+...+e−anx−e−b1x−e−b2x−...−e−bnxx(1−e−cx)dx=ln(Γ(a1c)Γ(a2c)⋯Γ(anc)Γ(b1c)Γ(b2c)⋯Γ(bnc))
∫∞0e−sx[e−a1x+e−a2x+...+e−anx−e−b1x−e−b2x−...−e−bnxx(1−e−cx)]dx=ln(Γ(a1+sc)Γ(a2+sc)⋯Γ(an+sc)Γ(b1+sc)Γ(b2+sc)⋯Γ(bn+sc))
Proof
We show the proof of the particular case, the generalization can be shown in a similar manner.
Consider the Laplace transform:
∫∞0e−sx[e−x−e−2x−e−3x+e−4xx(1−e−5x)]dx
Then
dds∫∞0e−sx[e−x−e−2x−e−3x+e−4xx(1−e−5x)]dx=−∫∞0e−sx[e−x−e−2x−e−3x+e−4x(1−e−5x)]dx
Hence
∫∞0e−sx[e−x−e−2x−e−3x+e−4x(1−e−5x)]dx=∫∞0e−sx∞∑n=0e−5nx(e−x−e−2x−e−3x+e−4x)dx=∞∑n=0∫∞0e−x(s+5n)(e−x−e−2x−e−3x+e−4x)dx=∞∑n=0∫∞0[e−x(s+5n+1)−e−x(s+5n+2)−e−x(s+5n+3)+e−x(s+5n+4)]dx=∞∑n=0[∫∞0e−x(s+5n+1)dx−∫∞0e−x(s+5n+2)dx−∫∞0e−x(s+5n+3)dx+∫∞0e−x(s+5n+4)dx]=∞∑n=0∫∞0[e−x(s+5n−1)−e−x(s+5n−2)−e−x(s+5n−3)+e−x(s+5n−4)]dx=∞∑n=0[1s+5n+1−1s+5n+2−1s+5n+3+1s+5n+4]
Then
∫∞0e−sx[e−x−e−2x−e−3x+e−4xx(1−e−5x)]dx=−∫∫∞0e−sx[e−x−e−2x−e−3x+e−4x(1−e−5x)]dxds+C=−∫∞∑n=0[1s+5n+1−1s+5n+2−1s+5n+3+1s+5n+4]ds+C=−∞∑n=0[∫1s+5n+1ds−∫1s+5n+2ds−∫1s+5n+3ds+∫1s+5n+4ds]+C=−∞∑n=0[ln(s+5n+1)−ln(s+5n+2)−ln(s+5n+3)+ln(s+5n+4)]+C=−∞∑n=0ln[(s+5n+1)(s+5n+4)(s+5n+2)(s+5n+3)]+C
Therefore
If s→0 ∫∞0e−x−e−2x−e−3x+e−4xx(1−e−5x)dx=−∞∑n=0ln[(5n+1)(5n+4)(5n+2)(5n+3)]+C
If s→∞
0=−lims→∞∞∑n=0ln[(s+5n+1)(s+5n+4)(s+5n+2)(s+5n+3)]+C=−∞∑n=0ln[lims→∞(s+5n+1)(s+5n+4)(s+5n+2)(s+5n+3)]+C=−∞∑n=0ln(1)+C⟹C=0
Then
∫∞0e−x−e−2x−e−3x+e−4xx(1−e−5x)dx=−∞∑n=0ln[(5n+1)(5n+4)(5n+2)(5n+4)]=−ln[∞∏n=0(5n+1)(5n+4)(5n+2)(5n+3)]
Recall the following infinite product representation for a quotient of products of Gamma functions (proof in the Appendix):
If m∑j=1aj=m∑j=1bj then: ∞∏k=0(a1+k)(a2+k)⋯(am+k)(b1+k)(b2+k)⋯(bm+k)=Γ(b1)Γ(b2)⋯Γ(bm)Γ(a1)Γ(a2)⋯Γ(am)
Hence, we can write:
∞∏n=0(5n+1)(5n+4)(5n+2)(5n+3)=∞∏n=0(n+15)(n+45)(n+25)(n+35)=Γ(25)Γ(35)Γ(15)Γ(45)=21+√5
Therefore
∫∞0e−x−e−2x−e−3x+e−4xx(1−e−5x)dx=ln(1+√52)
As a corollary we have the Laplace transform:
∫∞0e−sx[e−x−e−2x−e−3x+e−4xx(1−e−5x)]dx=ln[Γ(1+s5)Γ(4+s5)Γ(2+s5)Γ(3+s5)]
Appendix: Euler's infinite product representation for a quotient of Gamma functions
Recall the Euler's definition of the Gamma function Γ(z)=limn→∞(n+1)zn!z(z+1)(z+2)⋯(z+n)
Supposse that m∑j=1aj=m∑j=1bj
Hence
Γ(b1)Γ(b2)⋯Γ(bm)Γ(a1)Γ(a2)⋯Γ(am)=limn→∞m∏j=1[(n+1)bj−ajn∏k=0aj+kbj+k]=limn→∞(n+1)∑mj=1aj−∑mj=1bjm∏j=1n∏k=0aj+kbj+k=limn→∞n∏k=0m∏j=1aj+kbj+k
Therefore:
∞∏k=0(a1+k)(a2+k)⋯(am+k)(b1+k)(b2+k)⋯(bm+k)=Γ(b1)Γ(b2)⋯Γ(bm)Γ(a1)Γ(a2)⋯Γ(am) if m∑j=1aj=m∑j=1bj
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