Editing Legendre polynomials (section)

=== Zeros ===
All <math> n</math> zeros of <math>P_n(x)</math> are real, distinct from each other, and lie in the interval <math>(-1,1)</math>. Furthermore, if we regard them as dividing the interval <math>[-1,1]</math> into <math> n+1 </math> subintervals, each subinterval will contain exactly one zero of <math>P_{n+1}</math>. This is known as the interlacing property. Because of the parity property it is evident that if <math>x_k</math> is a zero of <math>P_n(x)</math>, so is <math>-x_k</math>. These zeros play an important role in [[numerical integration]] based on [[Gaussian quadrature]]. The specific quadrature based on the <math>P_n</math>'s is known as [[Gauss-Legendre quadrature]].

The zeros of <math>P_n(\cos \theta)</math> are distributed nearly uniformly over the range of <math>\theta \in (0, \pi)</math>, in the sense that there is one zero <math>\theta \in \left(\frac{\pi(k + 1/2)}{n + 1/2}, \frac{\pi(k + 1)}{n + 1/2}\right)</math> per <math>k = 0, 1, \dots, n-1</math>.<ref>{{Cite journal |last=Askey |first=Richard |date=November 1969 |title=Mehler's Integral for P_n (cos θ) |url=https://www.tandfonline.com/doi/abs/10.1080/00029890.1969.12000407 |journal=The American Mathematical Monthly |language=en |volume=76 |issue=9 |pages=1046–1049 |doi=10.1080/00029890.1969.12000407 |issn=0002-9890}}</ref> This can be proved by looking at the first formula of Dirichlet-Mehler.<ref>{{Cite journal |last=Bruns |first=H. |date=1881 |title=Zur Theorie der Kugelfunctionen. |url=https://www.degruyter.com/document/doi/10.1515/crll.1881.90.322/html |journal=CRLL |language=en |volume=1881 |issue=90 |pages=322–328 |doi=10.1515/crll.1881.90.322 |issn=1435-5345}}</ref>

From this property and the facts that <math> P_n(\pm 1) \ne 0 </math>, it follows that <math> P_n(x) </math> has <math> n-1 </math> local minima and maxima in <math> (-1,1) </math>. Equivalently, <math> dP_n(x)/dx </math> has <math> n -1 </math> zeros in <math> (-1,1) </math>.