### The Maxima on-line user's manual

Algebra Calculator

#### Li

Function: li [<s>] (<z>) Represents the polylogarithm function of order <s> and argument <z>, defined by the infinite series

inf ==== k \ z Li (z) = > -- s / s ==== k k = 1

`li [1]` is `- log (1 - z)`. `li [2]` and `li [3]` are the dilogarithm and trilogarithm functions, respectively.

When the order is 1, the polylogarithm simplifies to `- log (1 - z)`, which in turn simplifies to a numerical value if <z> is a real or complex floating point number or the `numer` evaluation flag is present.

When the order is 2 or 3, the polylogarithm simplifies to a numerical value if <z> is a real floating point number or the `numer` evaluation flag is present.

Examples:

```          (%i1) assume (x > 0);
(%o1)                        [x > 0]
(%i2) integrate ((log (1 - t)) / t, t, 0, x);
(%o2)                       - li (x)
2
(%i3) li [2] (7);
(%o3)                        li (7)
2
(%i4) li [2] (7), numer;
(%o4)        1.24827317833392 - 6.113257021832577 %i
(%i5) li [3] (7);
(%o5)                        li (7)
3
(%i6) li [2] (7), numer;
(%o6)        1.24827317833392 - 6.113257021832577 %i
(%i7) L : makelist (i / 4.0, i, 0, 8);
(%o7)   [0.0, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0]
(%i8) map (lambda ([x], li [2] (x)), L);
(%o8) [0, .2676526384986274, .5822405249432515,
.9784693966661848, 1.64493407, 2.190177004178597
- .7010261407036192 %i, 2.374395264042415
- 1.273806203464065 %i, 2.448686757245154
- 1.758084846201883 %i, 2.467401098097648
- 2.177586087815347 %i]
(%i9) map (lambda ([x], li [3] (x)), L);
(%o9) [0, .2584613953442624, 0.537213192678042,
.8444258046482203, 1.2020569, 1.642866878950322
- .07821473130035025 %i, 2.060877505514697
- .2582419849982037 %i, 2.433418896388322
- .4919260182322965 %i, 2.762071904015935
- .7546938285978846 %i]```

There are also some inexact matches for `li`. Try `?? li` to see them.

```(%o1)                                true
(%i2) ```

### Related Examples

##### li-makelist

fli:closed_loop(makel...

Calculate

##### li-sin-sqrt

li:[sin(%pi/7),sqrt(2...

li+[a,b,c];

sqrt(li);

Calculate

li(95);

Calculate

##### li-solve

li:(z0*v0*cp)/((z0/zl...

eq1: tp^2=li*z0/v0*(l...

solve(eq1, zl);

Calculate

##### li-solve

li:(z0*v0*cp)/((z0/zl...

eq1: tp^2=li*z0/v0*(l...

solve(eq1, zl);

Calculate

##### li-makelist

fli:closed_loop(makel...

root_locus(fli,xrange...

Calculate

##### li-ratsimp-solve

li:(z0*v0*cp)/((z0/zl...

eq1: tp^2=li*z0/v0*(l...

ratsimp(solve(eq1, zl));

Calculate

##### li-makelist

fli:closed_loop(makel...

root_locus(fli,xrange...

Calculate

##### li-ratsimp-solve

li:(z0*v0*cp)/((z0/zl...

eq1: tp^2=li*z0/v0*(l...

ratsimp(solve(eq1, zl));

Calculate

li(97.);

Calculate