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. »
Sinc can be used with Interval and CenteredInterval objects. »

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 (43)

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:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix Sinc function using MatrixFunction:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

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 (10)

Sinc is defined for all real and complex values:

Approximate real range of Sinc:

Sinc is an even function:

Sinc is an analytic function of x:

Sinc is monotonic in a specific range:

Sinc is not injective:

Not surjective:

Sinc is neither non-negative nor non-positive:

Sinc has no singularities or discontinuities:

Sinc is neither convex nor concave:

In [-π/2, π/2], it is concave:

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)

Non‐trivial minima and maxima of Sinc do not have ordinary closed forms:

Find numerical approximations:

Neat Examples (1)

A surprising sequence:

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