DiscreteDelta

The Kronecker-delta-like discrete delta: 1 when every argument is exactly 0, 0 otherwise.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageDiscreteDelta!↗

DiscreteDelta(x1, x2, …)1 if every argument is exactly 0; 0 if any argument is a nonzero real; unevaluated if any argument is undecided and none is a nonzero real.

Typed signatures

Type	Package	Applies
(real, real*) -> number	analytic	always

Domain: Elementary functions

Examples

Basic #

live ↗

In$\mathrm{DiscreteDelta}(0)$Out$1$

live ↗

In$\mathrm{DiscreteDelta}(1)$Out$0$

Scope #

Multi-argument -- 1 only when every argument is 0

live ↗

In$\mathrm{DiscreteDelta}(0, 0)$Out$1$

live ↗

In$\mathrm{DiscreteDelta}(1, 0)$Out$0$

Equals 1 when every argument is 0, however many there are

live ↗

In$\mathrm{DiscreteDelta}(0, 0, 0)$Out$1$

Four zero arguments still give 1

live ↗

In$\mathrm{DiscreteDelta}(0, 0, 0, 0)$Out$1$

Sign assumptions on the arguments do not decide whether they are all zero, so it stays

live ↗

In$\mathrm{Assuming}(x\lt0∧y\gt0, \mathrm{Refine}(\mathrm{DiscreteDelta}(x, x, y)))$Out$\mathrm{DiscreteDelta}(x, x, y)$

live ↗

In$\mathrm{FullSimplify}(\mathrm{MatrixFunction}(\mathrm{DiscreteDelta}, \bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack5, 1\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack0, 0\bigr\rbrack, \bigl\lbrack5\mathrm{DiscreteDelta}^{\prime}(1), 0\bigr\rbrack\bigr\rbrack$

live ↗

In$\mathrm{DiscreteDelta}(e, \operatorname{G}, \frac{1}{2})$Out$0$

live ↗

In$\mathrm{FullSimplify}(\operatorname{Series}(\mathrm{DiscreteDelta}(\mathrm{x1}, \mathrm{x2}), \mathrm{x1}, \mathrm{x0}, 2))$Out$\mathrm{DiscreteDelta}(\mathrm{x0}, \mathrm{x2})+(\mathrm{x1}-\mathrm{x0})\frac{∂}{∂\mathrm{x0}} \mathrm{DiscreteDelta}(\mathrm{x0}, \mathrm{x2})+\frac{1}{2}(\frac{∂^{2}}{∂\mathrm{x0}^{2}} \mathrm{DiscreteDelta}(\mathrm{x0}, \mathrm{x2})(\mathrm{x1}-\mathrm{x0})^2)+O\left((\mathrm{x1}-\mathrm{x0})^3\right)$

Possible issues #

The argument is exactly $0$ by a trigonometric identity, but its double is $2.5\times10^{-17}$: it stays unevaluated rather than guess, as Wolfram leaves it

live ↗

In$\mathrm{DiscreteDelta}(\frac{1}{2}(\sqrt{2-(2+2^{1/2})^{1/2}})-\sin(\frac{π}{16}))$Out$\mathrm{DiscreteDelta}(\frac{1}{2}(\sqrt{2-(2+2^{1/2})^{1/2}})-\sin(\frac{π}{16}))$

About

Examples	11
Checked against	Wolfram Language 8 agree2 differ1 unevaluated
Crosswalk	Wolfram Language