Why is the polylogarithm equation for n=1 a logarithm?

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  • #1
MathematicalPhysicist
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the equation of pl is:
(infinity)
y= sum x^i/i^n
i=1
in here http://www.2dcurves.com/exponential/exponentialpo.html it states in the case n=1 it is a logarithm i want to know why?

for all of my knowledge it should be y=x/1+x^2/2+x^3/3+...
is this the taylor expansion series for a logarithm?
 
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  • #2
Not exactly. For one thing that series gives y= 0 when x= 0 and log(x) is not defined for x=0.

The series you give is the Taylor's series for ln(x+1).
 
  • #3
so is this what meant in the webpage?
 

1. What is a polylogarithm?

A polylogarithm is a mathematical function that is defined as the sum of powers of a given number raised to various exponents. It is a generalization of the logarithm function and is denoted by Lis(x), where s is the order of the polylogarithm and x is the argument.

2. What is the difference between a polylogarithm and a logarithm?

The main difference between a polylogarithm and a logarithm is that a polylogarithm can take on a non-integer value for its exponent, while a logarithm always has a whole number exponent. Additionally, the argument for a polylogarithm can also be a complex number, while the argument for a logarithm is usually restricted to positive real numbers.

3. What are some applications of polylogarithms?

Polylogarithms have various applications in mathematics, physics, and engineering. They are used in the study of special functions, number theory, and complex analysis. In physics, polylogarithms appear in the calculation of quantum mechanical amplitudes and in the study of Bose-Einstein condensates. They are also used in signal processing and coding theory in engineering.

4. How are polylogarithms calculated?

The calculation of polylogarithms can be done using various methods, including numerical methods and series expansions. One common method is the Euler-Maclaurin summation formula, which allows for the calculation of polylogarithms with non-integer values of the exponent. There are also special algorithms and software programs available for computing polylogarithms.

5. What is the Riemann zeta function and how is it related to polylogarithms?

The Riemann zeta function, denoted by ζ(s), is a special case of the polylogarithm function when the argument is 1. It is defined as the infinite sum of the reciprocals of all positive integers raised to the power of s. This function has important connections to number theory and is also used in the study of the distribution of prime numbers. Additionally, the Riemann zeta function can be expressed in terms of polylogarithms of different orders, making it a useful tool in understanding the properties of polylogarithms.

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