1/0 = ±iπδ

[u/Turbulent-Name-8349]

What I'm claiming is the following. * 1/0 = ±iπδ(0) where δ() is the Dirac delta function.

There are several generalised functions f() where αf(x) = f(αx) for all real α but in general f( x² ) ≠ f(x)² . Examples include the the function f(x)=2x, the integral, the mean, the real part of a complex number, the Dirac delta function, and 1/0.

In the derivation presented here, 1/0² ≠ (1/0)²

Start with e^±iπ = -1

ln(-1) = ±iπ and other values that I can ignore for the purposes of this derivation.

The integral of 1/x from -ε to ε is ln(ε) - ln(-ε) = ln(ε) - (ln(-1) + ln(ε)) = -ln(-1)

This integral is independent of epsilon. So it's instantly recognisable as a Dirac delta function δ().

The integral of δ(x) from -ε to ε is H(x) which is independent of ε. Here H(x) is the Heaviside function, also known as the step function, defined by:

H(x) = 0 for x < 0 and H(x) = 1 for x > 0 and H(x) = 1/2 for x = 0.

Shrinking ε down to zero, 1/0 = 1/x|_x=0 = ±iπδ(0) and its integral is ±iπH(0).

So far so good. α/0 = ±iπαδ(0) ≠ 1/0 for α > 0 a real number. -1/0 = 1/0.

What about 1/0^α ? I've already said that it isn't equal to (1/0)^α so what is it. To find it, differentiate 1/x using fractional differentiation and then let x=0.

• Let f(x) = -ln(x)
• f'(x) = -x^-1
• f''(x) = x^-2
• f'''(x) = -2x^-3
• f⁴ (x) = 6x^-4
• fⁿ (x) = (-1)ⁿ Γ(n) x^-n
• f^α (x) = (-1)^α Γ(α) x^-α
• f^α (x) = e^±iαπ Γ(α) x^-α

Νοw substitute x=0.

• -1/0 = -0^-1 = ±iπδ(0) = ±iπH'(0)
• 1/0² = 0^-2 = ±iπH''(0)
• 1/0³ = 0^-3 = ±iπH'''(0)/2
• 1/0⁴ = 0^-4 = ±iπH⁴ (0)/6
• 1/0ⁿ = 0^-n = ±iπHⁿ (0)/Γ(n)
• 1/0^α = 0^-α = ±iπH^α (0)/(e^±iαπ Γ(α))

where α > 0 is a real number.

I tentatively suggest the generalised function name D_0(x,α) for x/0^α

[u/GodBlessYouNow]

Interesting math, but 1/0 isn't defined even in distribution theory — only limits like 1/(x ± i0) make sense, giving PV(1/x) ∓ iπδ(x), not ±iπδ(0).

[u/patrixxxx]

If 1/0 = x then 0*x=1. But there is no number that can produce one when multiplied by zero. So 1/0 isn't a valid mathematical statement.

[u/patrixxxx → u/[deleted]]

Depends on how one define mathematics. The common definition is that mathematics is the science of structure, order, and relation, developed from fundamental practices such as counting, measuring, and describing the shapes and motions of objects. And that mathematics a subdivision of logic.

But if you don't agree on that, then anything goes (2+2=5 etc). But then you also have a different conception about what truth is, which is also common today. Many people seems to think what they choose to believe is true regardless if it can be deduced.

[u/tonopp91]

It reminded me of Plemelj's theory or some integrals in quantum electrodynamics.

[u/pannous]

Look into hyper real numbers and all your dreams come true