logint

Logarithmic integral function

Description

example

A = logint(x) evaluates the logarithmic integral function (integral logarithm).

Examples

Integral Logarithm for Numeric and Symbolic Arguments

logint returns floating-point or exact symbolic results depending on the arguments you use.

Compute integral logarithms for these numbers. Because these numbers are not symbolic objects, logint returns floating-point results.

A = logint([-1, 0, 1/4, 1/2, 1, 2, 10])
A =
   0.0737 + 3.4227i   0.0000 + 0.0000i  -0.1187 + 0.0000i  -0.3787 + 0.0000i...
     -Inf + 0.0000i   1.0452 + 0.0000i   6.1656 + 0.0000i

Compute integral logarithms for the numbers converted to symbolic objects. For many symbolic (exact) numbers, logint returns unresolved symbolic calls.

symA = logint(sym([-1, 0, 1/4, 1/2, 1, 2, 10]))
symA =
[ logint(-1), 0, logint(1/4), logint(1/2), -Inf, logint(2), logint(10)]

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

A = vpa(symA)
A =
[ 0.07366791204642548599010096523015...
 + 3.4227333787773627895923750617977i,...
0,...
-0.11866205644712310530509570647204,...
-0.37867104306108797672720718463656,...
-Inf,...
1.0451637801174927848445888891946,...
6.1655995047872979375229817526695]

Plot Integral Logarithm

Plot the integral logarithm function on the interval from 0 to 10.

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

Handle Expressions Containing Integral Logarithm

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

Find the first and second derivatives of the integral logarithm:

syms x
dA = diff(logint(x), x)
dA = diff(logint(x), x, x)
dA =
1/log(x)
 
dA =
-1/(x*log(x)^2)

Find the right and left limits of this expression involving logint:

A_r = limit(exp(1/x)/logint(x + 1), x, 0, 'right')
A_r =
Inf
A_l = limit(exp(1/x)/logint(x + 1), x, 0, 'left')
A_l =
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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Logarithmic Integral Function

The logarithmic integral function, also called the integral logarithm, is defined as follows:

logint(x)=li(x)=0x1ln(t)dt

Tips

  • logint(sym(0)) returns 1.

  • logint(sym(1)) returns -Inf.

  • logint(z) = ei(log(z)) for all complex z.

References

[1] Gautschi, W., and W. F. Cahill. “Exponential Integral and Related 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