Dilogarithm

The real and imaginary parts of the function

Dilogarithm is a special function in mathematics, which is denoted and is a special case of (n = 2) polylogarithm . Dogarithm is defined as

The above definition of the dilogarithm is true for complex values ​​of the variable z . For real values ​​of z = x , this function has a cut along the real axis from 1 to . Usually the value of the function on the section is determined so that the imaginary part of the dilo-log is negative:

Function often referred to as Euler's dilogarithm, in honor of Leonard Euler, who considered this function in 1768 [1] . Sometimes the dilogarithm is called the Spence's function , in honor of the Scottish mathematician William Spence ( William Spence , 1777–1815) [2] , who in the early nineteenth century investigated the functions corresponding to and . The name "dilogarithm" was introduced by Hill ( CJ Hill ) in 1828.

Content

• 1 Functional relationships
• 2 Partial values
• 3 Options associated with logarithm
• 4 Notes
• 5 References

Functional Relationships [edit]

For dilogarithm, there are a number of useful functional relationships,

For valid ,

Relations are also known that contain two independent variables — for example, the Hill identity:

Partial Values ​​[edit]

Using the relationship between the functions of x and 1 / x , we obtain

There are also a number of results for arguments related to the golden section. ,

as well as for the logarithm of an imaginary argument,

where G is the Catalan constant.

Relations for particular values

Functions related to the logarithm [edit]

• Clausen function

Occurs when considering a dilo-log, whose argument is on a unit circle in the complex plane,

In this way,

• Lobachevsky function

This function is used in the calculation of volumes in hyperbolic geometry, and it is associated with the Clausen function (and hence with the dilogarithm),

Sometimes a different definition of the Lobachevsky function is used,

• Integral arctangent

Occurs when considering the imaginary argument's logarithm,

In this way,

• Legendre function

This function is expressed in terms of logs as

In particular, .