You get a bonus - 1 coin for daily activity. Now you have 1 coin

Dilogarithm

Lecture




Dilogarithm

The real and imaginary parts of the function Dilogarithm

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

Dilogarithm

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 Dilogarithm . Usually the value of the function on the section is determined so that the imaginary part of the dilo-log is negative:

Dilogarithm

Function Dilogarithm 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 Dilogarithm and Dilogarithm . 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,

Dilogarithm

Dilogarithm

Dilogarithm

Dilogarithm

Dilogarithm

Dilogarithm

For valid Dilogarithm ,

Dilogarithm

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

Dilogarithm

Partial Values ​​[edit]

Dilogarithm

Dilogarithm

Dilogarithm

Dilogarithm

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

Dilogarithm

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

Dilogarithm

Dilogarithm

Dilogarithm

Dilogarithm

as well as for the logarithm of an imaginary argument,

Dilogarithm

where G is the Catalan constant.

Relations for particular values

Dilogarithm

Dilogarithm

Dilogarithm

Dilogarithm

Dilogarithm

Dilogarithm

Functions related to the logarithm [edit]

  • Clausen function Dilogarithm

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

Dilogarithm

In this way,

Dilogarithm

  • 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),

Dilogarithm

Sometimes a different definition of the Lobachevsky function is used,

Dilogarithm

  • Integral arctangent Dilogarithm

Occurs when considering the imaginary argument's logarithm,

Dilogarithm

In this way,

Dilogarithm

  • Legendre function Dilogarithm

This function is expressed in terms of logs as

Dilogarithm

In particular, Dilogarithm .

created: 2014-10-25
updated: 2021-04-07
132644



Rating 9 of 10. count vote: 2
Are you satisfied?:



Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Comprehensive analysis and operational calculus

Terms: Comprehensive analysis and operational calculus