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Rewriting the ratios of factorials in the radial part as products of binomials shows that the coefficients are integer numbers: = = () ().A notation as terminating Gaussian hypergeometric functions is useful to reveal recurrences, to demonstrate that they are special cases of Jacobi polynomials, to write down the differential equations, etc.:
A trigonometric polynomial can be considered a periodic function on the real line, with period some divisor of , or as a function on the unit circle. Trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm; [4] this is a special case of the Stone–Weierstrass theorem.
Their importance and role can be understood through the following example: the hypergeometric series 2 F 1 has the Legendre polynomials as a special case, and when considered in the form of spherical harmonics, these polynomials reflect, in a certain sense, the symmetry properties of the two-sphere or, equivalently, the rotations given by the ...
Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently): + = = (), which arise naturally in multipole expansions. The left-hand side of the equation is the generating function for the Legendre polynomials.
Therefore, in computer algebra, normal form is a weaker notion: A normal form is a representation such that zero is uniquely represented. This allows testing for equality by putting the difference of two objects in normal form. Canonical form can also mean a differential form that is defined in a natural (canonical) way.
To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots r 1 and r 2. To convert the standard form to vertex form, one needs a process called completing the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.
The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).
The examples above discuss the representation problem for the numbers 3 and 65 by the form + and for the number 1 by the form . We see that 65 is represented by x 2 + y 2 {\displaystyle x^{2}+y^{2}} in sixteen different ways, while 1 is represented by x 2 − 2 y 2 {\displaystyle x^{2}-2y^{2}} in infinitely many ways and 3 is not represented by ...