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Hi members,see attached PdF file.Have you any idea to prove thes integrals
Thank you
Thank you
The dilogarithm is a special function that is defined as the integral of the logarithm of a complex number over the unit circle in the complex plane. It is denoted as Li2(z) and is also known as the Spence's function.
The dilogarithm function is closely related to the natural logarithm. In fact, it can be expressed in terms of the natural logarithm as Li2(z) = -∫ln(1-z)/z dz. This means that the dilogarithm function is the antiderivative of the natural logarithm function.
Dilogarithms are useful in solving integrals because they can simplify complex integrands and provide closed-form solutions. They also have many applications in physics, number theory, and other areas of mathematics.
Yes, the dilogarithm function can be extended to negative and non-real values using analytic continuation. This means that the function can be defined in a larger domain beyond its original definition, which is limited to positive real numbers.
Yes, there are several special properties and identities associated with the dilogarithm function. These include the reflection formula Li2(1/z) = -Li2(z) - π²/6 - ln(z)ln(1-z), the relation to the Riemann zeta function ζ(2) = π²/6 = Li2(1), and the functional equation Li2(z) + Li2(1-z) = π²/6 - ln(z)ln(1-z).