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On the other hand, the polynomial f(x) = 0 is the identically zero function. It is the (trivial) constant function and every x is a root. Its graph is the x-axis in the plane. [3] Its graph is symmetric with respect to the y-axis, and therefore a constant function is an even function. [4]
If a real function has a domain that is self-symmetric with respect to the origin, it may be uniquely decomposed as the sum of an even and an odd function, which are called respectively the even part (or the even component) and the odd part (or the odd component) of the function, and are defined by = + (), and = ().
Even function: is symmetric with respect to the Y-axis. Formally, for each x: f (x) = f (−x). Odd function: is symmetric with respect to the origin. Formally, for each x: f (−x) = −f (x). Relative to a binary operation and an order: Subadditive function: for which the value of f (x + y) is less than or equal to f (x) + f (y).
When in addition m is even, the function is a polynomial. When m is zero and ℓ integer, these functions are identical to the Legendre polynomials. In general, when ℓ and m are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not polynomials when m is odd.
The derivative of a constant function is zero, as noted above, and the differential operator is a linear operator, so functions that only differ by a constant term have the same derivative. To acknowledge this, a constant of integration is added to an indefinite integral; this ensures that all possible solutions are included. The constant of ...
The Legendre polynomial is determined by the values used for the two constants and , where = if is odd and = if is even. [ 2 ] In the fourth representation, ⌊ n / 2 ⌋ {\displaystyle \lfloor n/2\rfloor } stands for the largest integer less than or equal to n / 2 {\displaystyle n/2} .
Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. [2] Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That ...
This directly results from the fact that the integrand e −t 2 is an even function (the antiderivative of an even function which is zero at the origin is an odd function and vice versa).