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Li

The Logarithmic Integral

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Li(x)

Parameters

x

-

expression

Description

• 

The logarithmic integral, Li(x), is defined as:

  

Lix=∫0x1lntⅆt,x0 

  

=Eilnx 

  

where the integral is to be understood as a Cauchy Principal Value integral.

• 

This definition is extended to complex arguments z via the formula Lix=Eilnx.  Note that the resulting branch cuts are the intervals ,0 and 0,1. However, since Lix is defined as a Cauchy principal value integral, the values on the branch cuts are "isolated". That is, the complex function Liz is not continuous onto the branch cuts from either above or below.

• 

Li(x) provides an approximation to the number of primes less than or equal to x.

Examples

Lix

Lix

(1)

Li10.

6.165599505

(2)

Li1000.

177.6096580

(3)

and the actual number of primes1000 is:

nopsselectisprime,`$`1..1000

168

(4)

convertLix,Ei

Eilnx

(5)

Li1.+I

0.6139116692+2.059584214I

(6)

Li0.5

0.3786710431

(7)

Li0.5+0.I

0.3786710431+3.141592654I

(8)

Li0.50.I

0.37867104313.141592654I

(9)

Li3.123

0.06158134361+4.063328884I

(10)

See Also

convert

Ei

initialfunctions

numtheory[pi]

 


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