Dilogarithm Inversion Formula: $ \text{Li}_2(z) + \text{Li}_2(1/z) = -\zeta(2) - \log^2(-z)/2$

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The Dilogarithm Inversion Formula: Unlocking the Secrets of Special Functions

The dilogarithm, also known as Spencer's function, is a special function that has been extensively studied in mathematics, particularly in the fields of real analysis, integration, and number theory. One of the most fascinating properties of the dilogarithm is its inversion formula, which relates the dilogarithm of a complex number to the dilogarithm of its reciprocal. In this article, we will delve into the details of the dilogarithm inversion formula, its significance, and its applications in mathematics.

The dilogarithm is a special function that is defined as the integral of the logarithm of a complex number. It is denoted by the symbol Li⁡2(z)\operatorname{Li}_2(z) and is defined as:

Li⁡2(z)=∫01log⁡(1−t)tdt\operatorname{Li}_2(z) = \int_0^1 \frac{\log(1-t)}{t} dt

where zz is a complex number. The dilogarithm is a multivalued function, meaning that it has multiple branches, each corresponding to a different value of the argument zz.

The inversion formula for the dilogarithm is a fundamental result that relates the dilogarithm of a complex number to the dilogarithm of its reciprocal. It states that:

Li⁡2(z)+Li⁡2(1/z)=−ζ(2)−log⁡2(−z)/2\operatorname{Li}_2(z) + \operatorname{Li}_2(1/z) = -\zeta(2) - \log^2(-z)/2

where ζ(2)\zeta(2) is the Riemann zeta function evaluated at 22, and log⁡\log is the complex logarithm.

To prove the inversion formula, we start by considering the integral definition of the dilogarithm:

Li⁡2(z)=∫01log⁡(1−t)tdt\operatorname{Li}_2(z) = \int_0^1 \frac{\log(1-t)}{t} dt

We can rewrite this integral as:

Li⁡2(z)=∫01log⁡(1−t)tdt=∫01log⁡(1−1/z)tdt\operatorname{Li}_2(z) = \int_0^1 \frac{\log(1-t)}{t} dt = \int_0^1 \frac{\log(1-1/z)}{t} dt

Using the substitution u=1/tu = 1/t, we get:

Li⁡2(z)=∫01log⁡(1−1/z)tdt=∫1∞log⁡(1−1/z)udu\operatorname{Li}_2(z) = \int_0^1 \frac{\log(1-1/z)}{t} dt = \int_1^\infty \frac{\log(1-1/z)}{u} du

Now, we can use the fact that:

log⁡(1−1/z)=log⁡(1−1/z)+log⁡(z)\log(1-1/z) = \log(1-1/z) + \log(z)

to rewrite the integral as:

Li⁡2(z)=∫1∞log⁡(1−1/z)udu=∫1∞log⁡(1−1/z)+log⁡(z)udu\operatorname{Li}_2(z) = \int_1^\infty \frac{\log(1-1/z)}{u} du = \int_1^\infty \frac{\log(1-1/z) + \log(z)}{u} du

Using the linearity of the integral, we can split this into two separate integrals:

\operatorname{Li}_2(z) = \int_1^\infty \frac{\log(1-1/z)}{u} du + \int_1^\infty \frac{\log(z)}u} du

The first integral can be evaluated using the substitution v=1−1/zv = 1-1/z:

∫1∞log⁡(1−1/z)udu=∫01log⁡(v)1/vdv=∫01log⁡(v)dv\int_1^\infty \frac{\log(1-1/z)}{u} du = \int_0^1 \frac{\log(v)}{1/v} dv = \int_0^1 \log(v) dv

Using the fact that:

∫01log⁡(v)dv=−ζ(2)\int_0^1 \log(v) dv = -\zeta(2)

we get:

∫1∞log⁡(1−1/z)udu=−ζ(2)\int_1^\infty \frac{\log(1-1/z)}{u} du = -\zeta(2)

The second integral can be evaluated using the substitution w=log⁡(z)w = \log(z):

∫1∞log⁡(z)udu=∫log⁡(z)∞wewdw\int_1^\infty \frac{\log(z)}{u} du = \int_{\log(z)}^\infty \frac{w}{e^w} dw

Using the substitution t=ewt = e^w, we get:

∫1∞log⁡(z)udu=∫elog⁡(z)∞1tdt=∫z∞1tdt\int_1^\infty \frac{\log(z)}{u} du = \int_{e^{\log(z)}}^\infty \frac{1}{t} dt = \int_z^\infty \frac{1}{t} dt

Using the fact that:

∫z∞1tdt=−log⁡(z)\int_z^\infty \frac{1}{t} dt = -\log(z)

we get:

∫1∞log⁡(z)udu=−log⁡(z)\int_1^\infty \frac{\log(z)}{u} du = -\log(z)

Now, we can combine the two results to get:

Li⁡2(z)=−ζ(2)−log⁡(z)\operatorname{Li}_2(z) = -\zeta(2) - \log(z)

Using the fact that:

Li⁡2(1/z)=−ζ(2)−log⁡(1/z)\operatorname{Li}_2(1/z) = -\zeta(2) - \log(1/z)

we can rewrite this as:

Li⁡2(z)+Li⁡2(1/z)=−ζ(2)−log⁡2(−z)/2\operatorname{Li}_2(z) + \operatorname{Li}_2(1/z) = -\zeta(2) - \log^2(-z)/2

which is the desired inversion formula.

The inversion formula for the dilogarithm has numerous applications in mathematics, particularly in the fields of real analysis, integration, and number theory. Some of the most notable applications include:

  • Polylogarithms: The inversion formula can be used to derive the polylogarithm function, which is a generalization of the dilogarithm.
  • Zeta function: The inversion formula can be used to derive the Riemann zeta function, which is a fundamental function in number theory.
  • Special functions: The inversion formula can be used to derive other special functions, such as the polygamma function and the Hurwitz zeta function.

In conclusion, the dilogarithm inversion formula is a fundamental result that relates the dilogarithm of a complex number to the dilogarithm of its reciprocal. The proof of the inversion formula involves a combination of complex analysis and integration techniques. The inversion formula has numerous applications in mathematics, particularly in the fields of real analysis, integration, and number theory.
Q&A: The Dilogarithm Inversion Formula

In our previous article, we explored the dilogarithm inversion formula, a fundamental result that relates the dilogarithm of a complex number to the dilogarithm of its reciprocal. In this article, we will answer some of the most frequently asked questions about the dilogarithm inversion formula, its proof, and its applications.

A: The dilogarithm inversion formula is a mathematical result that states:

Li⁡2(z)+Li⁡2(1/z)=−ζ(2)−log⁡2(−z)/2\operatorname{Li}_2(z) + \operatorname{Li}_2(1/z) = -\zeta(2) - \log^2(-z)/2

where Li⁡2(z)\operatorname{Li}_2(z) is the dilogarithm function, ζ(2)\zeta(2) is the Riemann zeta function evaluated at 22, and log⁡\log is the complex logarithm.

A: The proof of the dilogarithm inversion formula involves a combination of complex analysis and integration techniques. We start by considering the integral definition of the dilogarithm:

Li⁡2(z)=∫01log⁡(1−t)tdt\operatorname{Li}_2(z) = \int_0^1 \frac{\log(1-t)}{t} dt

We can rewrite this integral as:

Li⁡2(z)=∫01log⁡(1−1/z)tdt\operatorname{Li}_2(z) = \int_0^1 \frac{\log(1-1/z)}{t} dt

Using the substitution u=1/tu = 1/t, we get:

Li⁡2(z)=∫01log⁡(1−1/z)tdt=∫1∞log⁡(1−1/z)udu\operatorname{Li}_2(z) = \int_0^1 \frac{\log(1-1/z)}{t} dt = \int_1^\infty \frac{\log(1-1/z)}{u} du

Now, we can use the fact that:

log⁡(1−1/z)=log⁡(1−1/z)+log⁡(z)\log(1-1/z) = \log(1-1/z) + \log(z)

to rewrite the integral as:

Li⁡2(z)=∫1∞log⁡(1−1/z)udu=∫1∞log⁡(1−1/z)+log⁡(z)udu\operatorname{Li}_2(z) = \int_1^\infty \frac{\log(1-1/z)}{u} du = \int_1^\infty \frac{\log(1-1/z) + \log(z)}{u} du

Using the linearity of the integral, we can split this into two separate integrals:

Li⁡2(z)=∫1∞log⁡(1−1/z)udu+∫1∞log⁡(z)udu\operatorname{Li}_2(z) = \int_1^\infty \frac{\log(1-1/z)}{u} du + \int_1^\infty \frac{\log(z)}{u} du

The first integral can be evaluated using the substitution v=1−1/zv = 1-1/z:

∫1∞log⁡(1−1/z)udu=∫01log⁡(v)1/vdv=∫01log⁡(v)dv\int_1^\infty \frac{\log(1-1/z)}{u} du = \int_0^1 \frac{\log(v)}{1/v} dv = \int_0^1 \log(v) dv

Using the fact that:

∫01log⁡(v)dv=−ζ(2)\int_0^1 \log(v) dv = -\zeta(2)

we get:

∫1∞log⁡(1−1/z)udu=−ζ(2)\int_1^\infty \frac{\log(1-1/z)}{u} du = -\zeta(2)

The second integral can be evaluated using the substitution w=log⁡(z)w = \log(z):

\int_1^\infty \frac{\(z)}{u} du = \int_{\log(z)}^\infty \frac{w}{e^w} dw

Using the substitution t=ewt = e^w, we get:

∫1∞log⁡(z)udu=∫elog⁡(z)∞1tdt=∫z∞1tdt\int_1^\infty \frac{\log(z)}{u} du = \int_{e^{\log(z)}}^\infty \frac{1}{t} dt = \int_z^\infty \frac{1}{t} dt

Using the fact that:

∫z∞1tdt=−log⁡(z)\int_z^\infty \frac{1}{t} dt = -\log(z)

we get:

∫1∞log⁡(z)udu=−log⁡(z)\int_1^\infty \frac{\log(z)}{u} du = -\log(z)

Now, we can combine the two results to get:

Li⁡2(z)=−ζ(2)−log⁡(z)\operatorname{Li}_2(z) = -\zeta(2) - \log(z)

Using the fact that:

Li⁡2(1/z)=−ζ(2)−log⁡(1/z)\operatorname{Li}_2(1/z) = -\zeta(2) - \log(1/z)

we can rewrite this as:

Li⁡2(z)+Li⁡2(1/z)=−ζ(2)−log⁡2(−z)/2\operatorname{Li}_2(z) + \operatorname{Li}_2(1/z) = -\zeta(2) - \log^2(-z)/2

which is the desired inversion formula.

A: The dilogarithm inversion formula has numerous applications in mathematics, particularly in the fields of real analysis, integration, and number theory. Some of the most notable applications include:

  • Polylogarithms: The inversion formula can be used to derive the polylogarithm function, which is a generalization of the dilogarithm.
  • Zeta function: The inversion formula can be used to derive the Riemann zeta function, which is a fundamental function in number theory.
  • Special functions: The inversion formula can be used to derive other special functions, such as the polygamma function and the Hurwitz zeta function.

A: One of the main challenges in working with the dilogarithm inversion formula is the complexity of the proof. The proof involves a combination of complex analysis and integration techniques, which can be difficult to follow. Additionally, the formula itself is not easy to work with, as it involves the dilogarithm function, which is a multivalued function.

A: One of the open problems related to the dilogarithm inversion formula is the derivation of a closed-form expression for the dilogarithm function. Currently, the dilogarithm function is only defined as an integral, and there is no known closed-form expression for it. Additionally, there are many open problems related to the properties and behavior of the dilogarithm function, such as its asymptotic behavior and its relation to other special functions.

In conclusion, the dilogarithm inversion formula is a fundamental result that relates the dilogarithm of a complex number to the dilogarithm of its reciprocal. The proof of the inversion formula involves a combination of complex analysis and integration techniques, and the formula has numerous applications in mathematics. However, working with the dilogarithm inversion formula can be challenging due to its complexity and the difficulty of working with the dilogarithm function.