Sinc

Sinc[z]

gives .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The argument of Sinc is assumed to be in radians. (Multiply by Degree to convert from degrees.)
  • Sinc[z] is equivalent to Sin[z]/z for , but is 1 for .
  • For certain special arguments, Sinc automatically evaluates to exact values.
  • Sinc can be evaluated to arbitrary numerical precision.
  • Sinc automatically threads over lists.

Examples

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Basic Examples  (4)

The argument is given in radians:

Plot :

Plot over a subset of the complexes:

Find the Fourier transform of Sinc:

Scope  (37)

Numerical Evaluation  (6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex numbers:

Evaluate Sinc efficiently at high precision:

Sinc can be applied to real-valued intervals:

Sinc threads elementwise over lists and matrices:

Specific Values  (4)

Values at zero:

Values of Sinc at fixed points:

Values at infinity:

The zeros of Sinc:

Find the first positive zero using Solve:

Substitute in the result:

Visualize the result:

Visualization  (3)

Plot the Sinc function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

Function Properties  (4)

Sinc is defined for all real and complex values:

Approximate real range of Sinc:

Sinc is an even function:

TraditionalForm formatting:

Differentiation  (2)

First derivative:

Higher derivatives:

Integration  (3)

Indefinite integral of Sinc:

Verify the antiderivative:

Definite integral of Sinc:

More integrals:

Series Expansions  (4)

Taylor expansion for Sinc:

Plot the first three approximations for Sinc around :

General term in the series expansion of Sinc:

The first-order Fourier series:

Sinc can be applied to a power series:

Integral Transforms  (3)

Compute the Laplace transform using LaplaceTransform:

MellinTransform:

HankelTransform:

Function Identities and Simplifications  (4)

Definition of Sinc:

Sinc of a sum:

Expand assuming real variables x and y:

Convert to exponentials:

Function Representations  (4)

Representation through Bessel functions:

Representation through gamma function:

Representation in terms of MeijerG:

Sinc can be represented as a DifferentialRoot:

Applications  (3)

Single-slit diffraction pattern for a 4λ slit:

Sinc-filtered Cauchy distribution:

A sinc signal is unaltered by sinc filter:

Properties & Relations  (2)

Use FunctionExpand to expand expressions involving Sinc:

Use FullSimplify to simplify expressions involving Sinc:

Possible Issues  (1)

Nontrivial minima and maxima of Sinc do not have ordinary closed forms:

Find numerical approximations:

Neat Examples  (1)

A surprising sequence:

Introduced in 2007
 (6.0)