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

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The integral of the logarithmic functions over a unit interval, specifically $$\int^1_0 \frac{\log(1+x)\log(1-x)}{x}\,dx$$, is discussed with a focus on the correct method for multiplying series. The error identified in evinda's solution pertains to using the pointwise product instead of the Cauchy product for absolutely convergent power series. This distinction is crucial for accurately solving the integral. The conversation highlights the importance of proper series multiplication techniques in calculus. Understanding these methods is essential for resolving similar logarithmic integrals effectively.
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logarithm integral # (2)

Solve the following

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

$$\int_0^1 \frac{\log(1+x) \log(1-x)}{x} dx=\int_0^1 \sum_{n=1}^{\infty} (-1)^n \frac{x^{2n-1}}{n^2} dx=\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \left [ \frac{x^{2n}}{2n}\right ]_0^1=\frac{1}{2} \sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}$$
 
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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

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

$$\int^1_0 \frac{\log(1-x)\log(1+x)}{x}\,dx = -\zeta(2)\log(2)+\zeta(2)\log(2)- \frac{15}{24} \zeta(3)=\frac{5}{8}\zeta(3)$$
 
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