# Find the range of Legendre polynomials

The range of Legendre polynomials in the Reals domain is [-1, 1]. How can we calculate it using FunctionRange or other MMA code?

FunctionRange[{LegendreP[l, x], Element[x, Reals] && Abs[x] <= 1 && Element[l, Integers] && l > 0}, x, y]


(*Unable to find the range with the available methods. *)

• As the Legendre polynomials are polynomials, the function range in the reals extends to infinity. The function range in the interval [-1,1] is [-1,1] Feb 6 at 11:33
• For even degree $l$, the min over $[-1,1]$ is $\ge -1/2$ and increases with $l$. For odd degree, the min is $-1$. Feb 6 at 11:54
• FWIW: Unevaluated@LegendreP[l, Root[Function[x, (-1 - l) x LegendreP[l, x] + (1 + l) LegendreP[1 + l, x]], If[Mod[l, 2] == 0, 2, 1]]] <= y <= 1 /. l -> 112 // N[#, 16] & -- Throws an error without a numeric positive integer value for l, though. Feb 6 at 12:02
• FunctionRange[{LegendreP[24, x], RealAbs[x] <= 1}, x, y] outputs 1/4194304 (676039 - 202811700 \ Root[-676039 + 66927861 #^2 - 1940907969 #^4 + 25786348731 #\^6 - 189099890694 #^8 + 842354058546 #^10 - 2397469243554 #\^12 + 4452442880886 #^14 - 5369122297539 #^16 + 4050390505161 #\^18 - 1735881645069 #^20 + 322476036831 #^22& , 1, 0]^2 + 10039179150 \Root[-676039 + 66927861 #^2 - 1940907969 #^4 + 25786348731 #\^6 - 189099890694 #^8 + 842354058546 #^10 - 2397469243554 #\ ^12 + 4452442880886 #^14 - 5369122297539 #^16 + 4050390505161 #\^18 - ... Feb 6 at 16:45
• + 66927861 #^2 - 1940907969 #^4 + 25786348731 #\ ^6 - 189099890694 #^8 + 842354058546 #^10 - 2397469243554 #\ ^12 + 4452442880886 #^14 - 5369122297539 #^16 + 4050390505161 #\^18 - 1735881645069 #^20 + 322476036831 #^22& , 1, 0]^24) <= y <= 1 Feb 6 at 16:46

FunctionRange[{LegendreP[l, x], RealAbs[x] <= 1 && Element[l, Integers] && l > 0}, {x, l}, y, Method -> {"Reduced" -> True}]

• I'm afraid I don't see how this answers the question. In your own question statement, you said the domain was $[-1,]$, not $[-1,1] \times {\Bbb Z}^+$. You implied that $l$ was a parameter and explicitly stated $x$ was the only variable. Feb 7 at 17:27
• @Goofy In my question, both x and l are variables, and it's possible that a coding error in the question led to your misunderstanding. I did not intend to obtain an analytical expression for the minimum value of the Legendre polynomial over the interval [-1, 1] (where l is a parameter and x is a variable, and l is even), as this seems not very straightforward. However, for specific values assigned to l, such as l ->112, determining the range of the Legendre polynomial over [-1, 1]: FunctionRange[{LegendreP[112, x], RealAbs[x] <= 1.0}, x, y] (* -0.402798 <= y <= 1. *) Feb 8 at 2:26