Saturday, August 21, 2021

Laplace transform II

Another beautiful Laplace transform

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. 0exe2xe3x+e4xx(1e5x)dx=ln(1+52)
As a corollary we have the Laplace transform: 0esx[exe2xe3x+e4xx(1e5x)]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,biN{0}
Hence 0ea1x+ea2x+...+eanxeb1xeb2x...ebnxx(1ecx)dx=ln(Γ(a1c)Γ(a2c)Γ(anc)Γ(b1c)Γ(b2c)Γ(bnc))
0esx[ea1x+ea2x+...+eanxeb1xeb2x...ebnxx(1ecx)]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:
0esx[exe2xe3x+e4xx(1e5x)]dx
Then
dds0esx[exe2xe3x+e4xx(1e5x)]dx=0esx[exe2xe3x+e4x(1e5x)]dx
Hence
0esx[exe2xe3x+e4x(1e5x)]dx=0esxn=0e5nx(exe2xe3x+e4x)dx=n=00ex(s+5n)(exe2xe3x+e4x)dx=n=00[ex(s+5n+1)ex(s+5n+2)ex(s+5n+3)+ex(s+5n+4)]dx=n=0[0ex(s+5n+1)dx0ex(s+5n+2)dx0ex(s+5n+3)dx+0ex(s+5n+4)dx]=n=00[ex(s+5n1)ex(s+5n2)ex(s+5n3)+ex(s+5n4)]dx=n=0[1s+5n+11s+5n+21s+5n+3+1s+5n+4]
Then
0esx[exe2xe3x+e4xx(1e5x)]dx=0esx[exe2xe3x+e4x(1e5x)]dxds+C=n=0[1s+5n+11s+5n+21s+5n+3+1s+5n+4]ds+C=n=0[1s+5n+1ds1s+5n+2ds1s+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 s0 0exe2xe3x+e4xx(1e5x)dx=n=0ln[(5n+1)(5n+4)(5n+2)(5n+3)]+C
If s 0=limsn=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)+CC=0
Then
0exe2xe3x+e4xx(1e5x)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 mj=1aj=mj=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
0exe2xe3x+e4xx(1e5x)dx=ln(1+52)
As a corollary we have the Laplace transform:
0esx[exe2xe3x+e4xx(1e5x)]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 mj=1aj=mj=1bj Hence Γ(b1)Γ(b2)Γ(bm)Γ(a1)Γ(a2)Γ(am)=limnmj=1[(n+1)bjajnk=0aj+kbj+k]=limn(n+1)mj=1ajmj=1bjmj=1nk=0aj+kbj+k=limnnk=0mj=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 mj=1aj=mj=1bj

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