Editing Chebyshev polynomials (section)

===Trigonometric definition===

The Chebyshev polynomials of the first and second kind can be defined as the unique polynomials satisfying
<math display="block">T_n(\cos\theta) = \cos(n\theta)</math>
and
<math display="block">U_n(\cos\theta) = \frac{\sin\big((n + 1)\theta\big)}{\sin\theta},</math>
for {{math|1=''n'' = 0, 1, 2, 3, …}}.

An equivalent way to state this is via exponentiation of a [[complex number]]: given a complex number {{math|1=''z'' = ''a'' + ''bi''}} with absolute value of one,
<math display="block">z^n = T_n(a) + ib U_{n-1}(a).</math>
Chebyshev polynomials can be defined in this form when studying [[trigonometric polynomials]].<ref>{{Cite journal |last=Schaeffer |first=A. C. |date=1941 |title=Inequalities of A. Markoff and S. Bernstein for polynomials and related functions |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-47/issue-8/Inequalities-of-A-Markoff-and-S-Bernstein-for-polynomials-and/bams/1183503783.full |journal=Bulletin of the American Mathematical Society |volume=47 |issue=8 |pages=565–579 |doi=10.1090/S0002-9904-1941-07510-5 |issn=0002-9904|doi-access=free }}</ref>

That {{math|cos{{nnbsp}}''nx''}} is an {{mvar|n}}th-[[degree of a polynomial|degree]] polynomial in {{math|cos{{nnbsp}}''x''}} can be seen by observing that {{math|cos{{nnbsp}}''nx''}} is the [[complex number|real part]] of one side of [[de Moivre's formula]]:
<math display="block">\cos n \theta + i \sin n \theta = (\cos \theta + i \sin \theta)^n.</math>
The real part of the other side is a polynomial in {{math|cos{{nnbsp}}''x''}} and {{math|sin{{nnbsp}}''x''}}, in which all powers of {{math|sin{{nnbsp}}''x''}} are [[parity (mathematics)|even]] and thus replaceable through the identity {{math|1=cos<sup>2</sup>{{nnbsp}}''x'' + sin<sup>2</sup>{{nnbsp}}''x'' = 1}}. By the same reasoning, {{math|sin{{nnbsp}}''nx''}} is the [[complex number|imaginary part]] of the polynomial, in which all powers of {{math|sin{{nnbsp}}''x''}} are [[parity (mathematics)|odd]] and thus, if one factor of {{math|sin{{nnbsp}}''x''}} is factored out, the remaining factors can be replaced to create a {{math|(''n''&nbsp;−&nbsp;1)}}st-degree polynomial in {{math|cos{{nnbsp}}''x''}}.

For ''x'' outside the interval [-1,1], the above definition implies
<math display="block">T_n(x) = \begin{cases}
 \cos(n \arccos x)                          & \text{ if }~ |x| \le 1, \\
 \cosh(n \operatorname{arcosh} x)           & \text{ if }~ x \ge 1, \\ 
 (-1)^n \cosh(n \operatorname{arcosh}(-x) ) & \text{ if }~ x \le -1.
\end{cases}</math>