Legendre Polynomial Pₙ(x) Calculator

Enter a degree n and x to evaluate the Legendre polynomial Pₙ(x) via recurrence. Shows neighboring degrees, the derivative, the coefficient list, and a graph.

Input

Enter a degree n (a non-negative integer) and x to evaluate the Legendre polynomial Pₙ(x) via the recurrence relation.

Degree n

A non-negative integer (up to 200)

Value of x

Any real number; graphed over −1 to 1

Result

Value of P5(x)

0.08984375

at x = 0.5

Degree n

P5₋₁(x)

-0.2890625

P5₊₁(x)

0.32324219

P5′(x) derivative

-2.2265625

Graph of P5(x) over −1 to 1

The blue curve is P{n}(x); the orange dot marks your input x.

Coefficients of P5(x)

Coefficients for each power of x. Zero coefficients are omitted.

Power of x	Coefficient
5	7.875
3	-8.75
1	1.875

How it works

Starting from P0(x)=1 and P1(x)=x, the recurrence (n+1)P(n+1)(x)=(2n+1)x Pn(x) − n P(n-1)(x) is applied in order to obtain Pn(x).
On the interval −1 to 1 the value of Pn(x) always lies between −1 and 1 and has n zeros. The graph is drawn over this interval.
The derivative is computed from (x^2−1)Pn'(x)=n(x Pn(x) − P(n-1)(x)), with the formula Pn'(1)=n(n+1)/2 used at x=±1.
The coefficient list shows the coefficient of each power of x in the expansion of Pn(x), built from integer binomial coefficients. A closed form is combined with the recurrence to limit rounding error.

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