# Linearity of the Dirac delta function [closed]

It is known that the Dirac delta function is linear. I do not understand why Mathematica says that a DiracDelta[x] + b DiracDelta[x] is not equal to (a + b) DiracDelta[x], assuming a and b are real numbers larger than zero.

## closed as off-topic by m_goldberg, Henrik Schumacher, LCarvalho, rhermans, marchNov 15 '18 at 4:23

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• Try Collect[aDiracDelta[x] + bDiracDelta[x], DiracDelta[x]]. There are more serious problems with DiracDelta in Mathematica. – user64494 Nov 9 '18 at 5:07
• Integrate[(a DiracDelta[x] + b DiracDelta[x] - (a + b) DiracDelta[x]) \[Phi][ x], {x, -\[Infinity], \[Infinity]}] returns 0, though. – Henrik Schumacher Nov 9 '18 at 7:32
• a DiracDelta[x] + b DiracDelta[x] // Factor works. – Bill Watts Nov 9 '18 at 8:09

The expression

a DiracDelta[x] + b DiracDelta[x] == (a + b) DiracDelta[x]


is a tautology — basically just a statement of the distributive law of ordinary arthmetic. It is true for any function f, linear on not.

Reduce[a f[x] + b f[x] == (a + b) f[x]]


True

• That is great! Thank you very much! – MJM Nov 9 '18 at 17:12
• Please don't add "thank you" as an answer. Instead, accept the answer that you found most helpful. - From Review – Coolwater Nov 9 '18 at 17:35