MHB How Does One Solve the Integral of Logarithmic Functions Over a Unit Interval?

  • Thread starter Thread starter alyafey22
  • Start date Start date
  • Tags Tags
    Integral Logarithm
alyafey22
Gold Member
MHB
Messages
1,556
Reaction score
2
logarithm integral # (2)

Solve the following

$$\int^1_0 \frac{\log(1+x)\log(1-x)}{x}\,dx$$​
 
Last edited:
Mathematics news on Phys.org
ZaidAlyafey said:
Solve the following

$$\int^1_0 \frac{\log(1+x)\log(1-x)}{x}\,dx$$​

$$\log (1-x)=-\sum_{n=1}^{\infty} \frac{x^n}{n} \text{ for } |x|<1$$

$$\log (1+x)=\sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} \text{ for } |x|<1$$

So:

$$\frac{\log(1-x)\log(1+x)}{x}=\frac{- \sum_{n=1}^{\infty} \frac{x^n}{n} \cdot \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n}}{x}=\frac{\sum_{n=1}^{\infty} (-1)^{n+2} \frac{x^{2n}}{n^2}}{x}=\sum_{n=1}^{\infty} (-1)^n \frac{x^{2n-1}}{n^2}$$

Therefore:

$$\int_0^1 \frac{\log(1+x) \log(1-x)}{x} dx=\int_0^1 \sum_{n=1}^{\infty} (-1)^n \frac{x^{2n-1}}{n^2} dx=\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \left [ \frac{x^{2n}}{2n}\right ]_0^1=\frac{1}{2} \sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}$$
 
Last edited by a moderator:
evinda said:
$$\frac{\log(1-x)\log(1+x)}{x}=\frac{- \sum_{n=1}^{\infty} \frac{x^n}{n} \cdot \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n}}{x}=\frac{\sum_{n=1}^{\infty} (-1)^{n+2} \frac{x^{2n}}{n^2}}{x}=\sum_{n=1}^{\infty} (-1)^n \frac{x^{2n-1}}{n^2}$$

How did you perform the multiplication of the two series ?
 
In evinda's solution, there is an error made in multiplying the two series. The product of two absolutely convergent power series is given by the Cauchy product, not pointwise product.
 
Euge said:
In evinda's solution, there is an error made in multiplying the two series. The product of two absolutely convergent power series is given by the Cauchy product, not pointwise product.

I see..I am sorry! (Tmi)
 
$$\int^1_0 \frac{\log(1-x)\log(1+x)}{x}\,dx = -\zeta(2)\log(2)+\int^1_0 \frac{\mathrm{Li}_2(x)}{1+x}\,dx $$

$$\int^1_0 \frac{\mathrm{Li}_2(x)}{1+x}\,dx = \int^1_0 \sum_k (-x)^k \mathrm{Li}_2(x) = \sum_{k\geq 1} (-1)^{k-1}\int^1_0 x^{k-1} \mathrm{Li}_2(x) $$

Since

$$\int^1_0 x^{k-1}\mathrm{Li}_2(x)\, dx = \sum_{n\geq 1}\frac{1}{n^2(n+k)} =\frac{1}{k}\left(\zeta(2)-\frac{H_k}{k}\right)$$

We've got

$$\int^1_0 \frac{\mathrm{Li}_2(x)}{1+x}\,dx =\zeta(2)\sum_{k\geq 1} \frac{(-1)^{k-1}}{k}-\sum_{k\geq 1} (-1)^{k-1}\frac{H_k}{k^2} = \frac{1}{24}\left(4 \pi^2 \log(2)-15 \zeta(3)\right)$$

$$\int^1_0 \frac{\log(1-x)\log(1+x)}{x}\,dx = -\zeta(2)\log(2)+\zeta(2)\log(2)- \frac{15}{24} \zeta(3)=\frac{5}{8}\zeta(3)$$
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top