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dilog

Dilogarithm function

Syntax

Description

example

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

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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

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Dilogarithm Function

There are two common definitions of the dilogarithm function.

The implementation of the dilog function uses the following definition:

dilog(x)=1xln(t)1tdt

Another common definition of the dilogarithm function is

Li2(x)=x0ln(1t)tdt

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.

See Also

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Introduced in R2014a