The Polylogarithm Function

Oct 13, 2025 by windailys.com 21 views

Introduction

In this article, we will delve into the evaluation of a complex definite integral involving the polylogarithm function. The integral in question is $\int _0^{\pi /2}x\operatorname{Li}_3\left(\sin \left(x\right)\right)\:\mathrm{d}x$ . We will explore the properties of the polylogarithm function and its relation to other mathematical functions, and then proceed to evaluate the given integral.

The Polylogarithm Function

The polylogarithm function, denoted by $\operatorname{Li}_s(z)$ , is a complex function defined as:

\operatorname{Li}_s(z) = \sum_{k=1}^{\infty} \frac{z^k}{k^s}

This function is a generalization of the natural logarithm function, and it has been extensively studied in mathematics. The polylogarithm function has several interesting properties, including:

Analytic continuation: The polylogarithm function can be analytically continued to the entire complex plane, except for the branch points at $z=1$ and $z=-1$ .
Recurrence relations: The polylogarithm function satisfies several recurrence relations, which can be used to simplify and evaluate the function.
Connection to other functions: The polylogarithm function is connected to other mathematical functions, such as the Riemann zeta function, the Hurwitz zeta function, and the polygamma function.

The Integral in Question

The integral we are interested in evaluating is:

\int _0^{\pi /2}x\operatorname{Li}_3\left(\sin \left(x\right)\right)\:\mathrm{d}x

This integral involves the polylogarithm function with argument $\sin(x)$ , which is a periodic function with period $\pi$ . The integral is taken over the interval $[0, \pi/2]$ , which is a quarter of the period of the sine function.

Transformation of the Integral

To evaluate the integral, we can use a transformation to change the variable of integration. Let $x = \arcsin(u)$ , then $dx = \frac{1}{\sqrt{1-u^2}} du$ . Substituting these expressions into the integral, we get:

\int _0^{\pi /2}x\operatorname{Li}_3\left(\sin \left(x\right)\right)\:\mathrm{d}x = \int _0^1 \frac{\arcsin(u)\operatorname{Li}_3(u)}{\sqrt{1-u^2}} du

This transformation changes the variable of integration from $x$ to $u$ , and it also changes the limits of integration from $[0, \pi/2]$ to $[0, 1]$ .

Evaluation of the Integral

To evaluate the integral, we can use the properties of the polylogarithm function and the transformation we applied earlier. Using the recurrence relation for the polylogarithm function, we can simplify the integral as follows:

\int _0^1 \frac{\arcsin(u)\operatorname{Li}_3(u)}{\sqrt{1-u^2}} du = \int _0^1 \frac{\arcsin(u)\left(\frac{1}{2}\operatorname{Li}_2(u^2) + \frac{1}{6}\right)}{\sqrt{1-u^2}} du

This simplification uses the recurrence relation $\operatorname{Li}_3(z) = \frac{1}{2}\operatorname{Li}_2(z^2) + \frac{1}{6}$ .

Final Evaluation

\int _0^1 \frac{\arcsin(u)\left(\frac{1}{2}\operatorname{Li}_2(u^2) + \frac{1}{6}\right)}{\sqrt{1-u^2}} du = \frac{1}{2}\int _0^1 \frac{\arcsin(u)\operatorname{Li}_2(u^2)}{\sqrt{1-u^2}} du + \frac{1}{6}\int _0^1 \frac{\arcsin(u)}{\sqrt{1-u^2}} du

This simplification uses the recurrence relation $\operatorname{Li}_3(z) = \frac{1}{2}\operatorname{Li}_2(z^2) + \frac{1}{6}$ .

Evaluation of the First Integral

To evaluate the first integral, we can use the properties of the polylogarithm function and the transformation we applied earlier. Using the recurrence relation for the polylogarithm function, we can simplify the integral as follows:

\frac{1}{2}\int _0^1 \frac{\arcsin(u)\operatorname{Li}_2(u^2)}{\sqrt{1-u^2}} du = \frac{1}{2}\int _0^1 \frac{\arcsin(u)\left(\frac{1}{2}\operatorname{Li}_1(u^2) + \frac{1}{2}\right)}{\sqrt{1-u^2}} du

This simplification uses the recurrence relation $\operatorname{Li}_2(z) = \frac{1}{2}\operatorname{Li}_1(z^2) + \frac{1}{2}$ .

Evaluation of the Second Integral

To evaluate the second integral, we can use the properties of the polylogarithm function and the transformation we applied earlier. Using the recurrence relation for the polylogarithm function, we can simplify the integral as follows:

\frac{1}{6}\int _0^1 \frac{\arcsin(u)}{\sqrt{1-u^2}} du = \frac{1}{6}\int _0^1 \frac{\arcsin(u)}{\sqrt{1-u^2}} du

This simplification uses the fact that $\arcsin(u)$ is an odd function.

Final Answer

Using the properties of the polylogarithm function and the transformation we applied earlier, we can evaluate the integral as follows:

\int _0^{\pi /2}x\operatorname{Li}_3\left(\sin \left(x\right)\right)\:\mathrm{d}x = \frac{1}{2}\int _0^1 \frac{\arcsin(u)\operatorname{Li}_2(u^2)}{\sqrt{1-u^2}} du + \frac{1}{6}\int _0^1 \frac{\arcsin(u)}{\sqrt{1-u^2}} du

This evaluation uses the recurrence relations for the polylogarithm function and the transformation we applied earlier.

Conclusion

In this article, we evaluated the definite integral $\int _0^{\pi /2}x\operatorname{Li}_3\left(\sin \left(x\right)\right)\:\mathrm{d}x$ . We used the properties of the polylogarithm function and a transformation to change the variable of integration. We then applied the recurrence relations for the polylogarithm function to simplify the integral and evaluate it. The final answer is a combination of two integrals, which can be evaluated using the properties of the polylogarithm function and the transformation we applied earlier.

Introduction

In our previous article, we evaluated the definite integral $\int _0^{\pi /2}x\operatorname{Li}_3\left(\sin \left(x\right)\right)\:\mathrm{d}x$ . We used the properties of the polylogarithm function and a transformation to change the variable of integration. We then applied the recurrence relations for the polylogarithm function to simplify the integral and evaluate it. In this article, we will answer some common questions related to the evaluation of this integral.

Q: What is the polylogarithm function?

A: The polylogarithm function, denoted by $\operatorname{Li}_s(z)$ , is a complex function defined as:

\operatorname{Li}_s(z) = \sum_{k=1}^{\infty} \frac{z^k}{k^s}

This function is a generalization of the natural logarithm function, and it has been extensively studied in mathematics.

Q: What is the recurrence relation for the polylogarithm function?

A: The polylogarithm function satisfies several recurrence relations, which can be used to simplify and evaluate the function. One of the most useful recurrence relations is:

\operatorname{Li}_3(z) = \frac{1}{2}\operatorname{Li}_2(z^2) + \frac{1}{6}

This recurrence relation can be used to simplify the integral $\int _0^{\pi /2}x\operatorname{Li}_3\left(\sin \left(x\right)\right)\:\mathrm{d}x$ .

Q: How did you transform the integral?

A: We used a transformation to change the variable of integration from $x$ to $u$ , where $u = \sin(x)$ . This transformation changes the limits of integration from $[0, \pi/2]$ to $[0, 1]$ .

Q: How did you simplify the integral?

A: We used the recurrence relations for the polylogarithm function to simplify the integral. We applied the recurrence relation $\operatorname{Li}_3(z) = \frac{1}{2}\operatorname{Li}_2(z^2) + \frac{1}{6}$ to simplify the integral.

Q: What is the final answer?

A: The final answer is a combination of two integrals, which can be evaluated using the properties of the polylogarithm function and the transformation we applied earlier. The final answer is:

\int _0^{\pi /2}x\operatorname{Li}_3\left(\sin \left(x\right)\right)\:\mathrm{d}x = \frac{1}{2}\int _0^1 \frac{\arcsin(u)\operatorname{Li}_2(u^2)}{\sqrt{1-u^2}} du + \frac{1}{6}\int _0^1 \frac{\arcsin(u)}{\sqrt{1-u^2}} du

Q: What are some common applications of the polylogarithm function?

A: The polylogarithm function has several applications in mathematics and physics. Some common applications include:

Number theory: The polylogarithm function is used in number theory to study the properties of prime numbers and the distribution of prime numbers.
Algebraic geometry: The polylogarithm function is used in algebraic geometry to study the properties of algebraic curves and surfaces.
Physics: The polylogarithm function is used in physics to study the properties of particles and fields in quantum field theory.

Q: What are some common challenges when working with the polylogarithm function?

A: Some common challenges when working with the polylogarithm function include:

Convergence: The polylogarithm function is a divergent series, which can make it difficult to work with.
Branch points: The polylogarithm function has branch points at $z=1$ and $z=-1$ , which can make it difficult to work with.
Recurrence relations: The polylogarithm function satisfies several recurrence relations, which can make it difficult to simplify and evaluate the function.

Conclusion

In this article, we answered some common questions related to the evaluation of the definite integral $\int _0^{\pi /2}x\operatorname{Li}_3\left(\sin \left(x\right)\right)\:\mathrm{d}x$ . We discussed the properties of the polylogarithm function, the recurrence relations for the polylogarithm function, and the transformation we applied to change the variable of integration. We also discussed some common applications and challenges when working with the polylogarithm function.