Sinc¶

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description¶

Sinc[z]

gives the cardinal sine Sin[z]/z, with Sinc[0] = 1.

D[Sinc[z], z] = Cos[z]/z - Sin[z]/z^2. Listable.

Notes

An entire, even function. Sinc\[+-Infinity\] = 0. Real and complex inputs evaluate numerically at machine or arbitrary (MPFR) precision;

Examples (6)¶

Every input below was run against the current Mathilda build and its output recorded.

Applications (6)¶

In[1]:= Sinc[0]
Out[1]= 1

In[2]:= Sinc[2.]
Out[2]= 0.454649

In[3]:= N[Sinc[2], 45]
Out[3]= 0.454648713412840847698009932955872421351127485

In[4]:= Sinc[1. + I]
Out[4]= 0.966711 - 0.331747 I

In[5]:= D[Sinc[x], x]
Out[5]= Cos[x]/x - Sin[x]/x^2

In[6]:= Series[Sinc[x], {x, 0, 6}]
Out[6]= 1 - x^2/6 + x^4/120 - x^6/5040 + O[x]^7

Algorithm¶

 Mathilda -- the cardinal sine  Sinc[z] = Sin[z]/z  (Sinc[0] = 1).

Sinc is entire and even, with a removable singularity at the origin. Each kind of argument takes the cheapest route:

  exact special values   ->  1 (at 0), 0 (at +-Infinity), Indeterminate
  machine real           ->  libm sin(x)/x
  arbitrary real (MPFR)  ->  mpfr_sin(x)/x at the input precision
  complex (any prec)     ->  sin(z)/z via the shared ncpx toolkit
  everything else        ->  stays symbolic (return NULL)

Attributes: Listable, NumericFunction, Protected.

Implementation notes¶

Attributes: Listable, NumericFunction, Protected.

References¶

See also: SinIntegral

Source: src/info.c
Specification: docs/spec/builtins/special-functions.md
Tests: tests/test_compile.c
Tests: tests/test_interval.c
Tests: tests/test_numeric_stress.c
Tests: tests/test_sinc.c

Notes & additional examples¶

Notes¶

Sinc[z] is the cardinal sine Sin[z]/z, with the removable singularity at the origin filled in as Sinc[0] = 1. It is entire and even, and Sinc[±Infinity] = 0. It appears as the derivative of the sine integral: D[SinIntegral[z], z] = Sinc[z]. Numeric evaluation is at machine or arbitrary (MPFR) precision for both real and complex arguments. Listable. See also SinIntegral.