Another fun logarithm integral

[alyafey22]

Here is a fun exercise to prove

\(\displaystyle \int^1_0 \frac{\log(t) \log(1-t)}{t} \, dt = \zeta(3)\)

 

[alyafey22]

Here is a small hint

Spoiler

Try integration by parts

 

[alyafey22]

We can evaluate the definite integral

Integrating by parts we get the following

\(\displaystyle \int \frac{\log(x) \log(1-x)}{x}\, dx = -\text{Li}_2(x) \log(x) + \text{Li}_3(x) +C\)\(\displaystyle \int^1_0 \frac{\log(x) \log(1-x)}{x}\, dx = \text{Li}_3(1) = \zeta(3) \)

 

[chisigma]

Here is a fun exercise to prove

\(\displaystyle \int^1_0 \frac{\log(t) \log(1-t)}{t} \, dt = \zeta(3)\)

In...

http://www.mathhelpboards.com/f49/integrals-natural-logarithm-5286/

... is reported that if...

$$ f(x) = \sum_{k=0}^{\infty} a_{k}\ x^{k}\ (1)$$

... then...

$$\int_{0}^{1} f(x)\ \ln^{n} x\ d x = (-1)^{n} n!\ \sum_{k=0}^{\infty} \frac{a_{k}}{(k+1)^{n+1}}\ (2)$$

In this case is n=1 and $\displaystyle a_{k}= - \frac{1}{k+1}$, so that...

$$\int_{0}^{1} \frac{\ln (1-x)}{x}\ \ln x\ dx = \zeta (3)\ (3)$$

Kind regards

$\chi$ $\sigma$

 

[CellCoree]

I appreciate the use of mathematics to explore and understand natural phenomena. The integral presented is indeed a fascinating one, as it involves the natural logarithm function and the Riemann zeta function. The proof of this integral involves using the Taylor series expansion of the natural logarithm and manipulating it to match the series expansion of the Riemann zeta function. This requires a deep understanding of both functions and their properties, making it a challenging but rewarding exercise for mathematicians.

Furthermore, this integral has important applications in various fields of science, such as physics and engineering. For example, it can be used in the study of diffusion processes, where the logarithm function appears in the solution of the diffusion equation. Additionally, the Riemann zeta function has connections to the distribution of prime numbers, which has implications in number theory and cryptography.

Overall, this integral serves as an excellent example of the beauty and utility of mathematics in scientific research. It highlights the interconnectedness of different mathematical concepts and their applications in real-world problems. As scientists, it is important to continue exploring and understanding these connections to advance our knowledge and contribute to the development of new technologies and theories.