dilog

Dilogarithm function

Syntax

dilog(X)

Description

dilog(X) returns the dilogarithm function.

Examples

Dilogarithm Function for Numeric and Symbolic Arguments

Depending on its arguments, dilog returns floating-point or exact symbolic results.

Compute the dilogarithm function for these numbers. Because these numbers are not symbolic objects, dilog returns floating-point results.

A = dilog([-1, 0, 1/4, 1/2, 1, 2])

A =
   2.4674 - 2.1776i   1.6449 + 0.0000i   0.9785 + 0.0000i...
   0.5822 + 0.0000i   0.0000 + 0.0000i  -0.8225 + 0.0000i

Compute the dilogarithm function for the numbers converted to symbolic objects. For many symbolic (exact) numbers, dilog returns unresolved symbolic calls.

symA = dilog(sym([-1, 0, 1/4, 1/2, 1, 2]))

symA =
[ pi^2/4 - pi*log(2)*1i, pi^2/6, dilog(1/4), pi^2/12 - log(2)^2/2, 0, -pi^2/12]

Use vpa to approximate symbolic results with floating-point numbers:

vpa(symA)

ans =
[ 2.467401100272339654708622749969 - 2.1775860903036021305006888982376i,...
1.644934066848226436472415166646,...
0.97846939293030610374306666652456,...
0.58224052646501250590265632015968,...
0,...
-0.82246703342411321823620758332301]

Plot Dilogarithm Function

Plot the dilogarithm function on the interval from 0 to 10.

syms x
fplot(dilog(x),[0 10])
grid on

Handle Expressions Containing Dilogarithm Function

Many functions, such as diff, int, and limit, can handle expressions containing dilog.

Find the first and second derivatives of the dilogarithm function:

syms x
diff(dilog(x), x)
diff(dilog(x), x, x)

ans =
-log(x)/(x - 1)
 
ans =
log(x)/(x - 1)^2 - 1/(x*(x - 1))

Find the indefinite integral of the dilogarithm function:

int(dilog(x), x)

ans =
x*(dilog(x) + log(x) - 1) - dilog(x)

Find the limit of this expression involving dilog:

limit(dilog(x)/x, Inf)

ans =
0

Input Arguments

X — Input

symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix

Input, specified as a symbolic number, variable, expression, or function, or as a vector or matrix of symbolic numbers, variables, expressions, or functions.

More About

Dilogarithm Function

There are two common definitions of the dilogarithm function.

The implementation of the dilog function uses the following definition:

$$\text{dilog}\left(x\right)={\displaystyle \underset{1}{\overset{x}{\int }}\frac{\ln \left(t\right)}{1-t}}dt$$

Another common definition of the dilogarithm function is

$${\text{Li}}_2\left(x\right)={\displaystyle \underset{x}{\overset{0}{\int }}\frac{\ln \left(1-t\right)}{t}}dt$$

Thus, dilog(x) = Li₂(1 – x).

Tips

• dilog(sym(-1)) returns pi^2/4 - pi*log(2)*i.

• dilog(sym(0)) returns pi^2/6.

• dilog(sym(1/2)) returns pi^2/12 - log(2)^2/2.

• dilog(sym(1)) returns 0.

• dilog(sym(2)) returns -pi^2/12.

• dilog(sym(i)) returns pi^2/16 - (pi*log(2)*i)/4 - catalan*i.

• dilog(sym(-i)) returns catalan*i + (pi*log(2)*i)/4 + pi^2/16.

• dilog(sym(1 + i)) returns - catalan*i - pi^2/48.

• dilog(sym(1 - i)) returns catalan*i - pi^2/48.

• dilog(sym(Inf)) returns -Inf.

References

[1] Stegun, I. A. “Miscellaneous Functions.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.

Version History

Introduced in R2014a