sinpi

sinpi is a mathematical function defined by sinpi(x) = sin(π x), where π denotes pi. It is commonly used in mathematics and programming as a shorthand for the sine of pi times x. The function is periodic with period 2, since sin(π(x+2)) = sin(πx + 2π) = sin(πx).

Zeros occur at all integers: sinpi(n) = 0 for n ∈ Z. At half-integers, sinpi(n + 1/2) equals (−1)^n, giving the maximum and minimum values of ±1 at those points. The function is odd, satisfying sinpi(−x) = −sinpi(x), and its range is between −1 and 1.

The derivative is d/dx sinpi(x) = π cos(π x), and the second derivative is −π^2 sin(π x). These relationships reflect the standard trigonometric identity adapted to the π-scaled argument.

In complex analysis, sin(π z) extends to an entire function of a complex variable, with zeros at every integer. Sinpi as a real-valued function inherits this behavior through its definition.

Numerical and software usage: many programming environments implement sinpi as a separate function to improve numerical stability, by reducing the argument before evaluation of sine. This helps maintain accuracy for large or highly precise inputs. It is available in several environments, including MATLAB/Octave, Julia, and other scientific computing libraries, where sinpi(x) is equivalent to sin(π x).

Sinpi is distinct from the sinc function, which is sin(x)/x. Sinpi focuses on the sine of π times the input, playing a role in various analytic and computational contexts where π-periodic behavior is relevant.