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More On The Dirac Delta Function

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Posted by Alan on June 24, 2002 02:01:06 UTC

I have found a section in a textbook, about the Dirac delta function.

It appears to show that Dr. Stafford is correct.

In the following, I use the word "by" to denote where a math symbol is next to another (like if the integral symbol is next in a line of symbols, I write "by the integral...")

Page 207 leads to two defining relationships for the Fourier transform

f(t) = 1 divided by: (2pi) to power of a half; by the integral from minus infinity to plus infinity g(omega) exp (i omega t) d omega

(8.31)

and

g(omega) = 1 divided by: (2pi) to power of a half; by the integral from minus infinity to plus infinity f(t) exp (-1 omega t) d t

(8.32)

Now, the book refers to these earlier defining functions which, substituting one for the other gives the equation

f(x) = (2pi) to the power of: minus a half,
by the integral from -infinity to plus infinity,
times d omega exp (i omega t),
multiplied by (2pi) to the power minus a half, by the integral from minus infinity to plus infinity, by d t dash exp (-1 omega t dash) f(t dash)

EQUALS

the integral from minus infinty to plus infinity , by d t dash f( t dash) x (2pi) to power of minus 1, by the integral from minus infinity to plus infinity, by d omega exp

(Equation 8.46)

It says: "Here we have written the differentials immediately following the integral signs to which they refer and in obtaining the second line from the first we have assumed that the order of the integrations can be reversed.

Now, if it is recalled that f(t) is an arbitrary function, and also noted that in 8.46 the left-hand side refers to a value of f at a PARTICULAR value of t, whilst the right-hand side contains an integral over ALL values of the argument of f, then it is clear that the expression

(2pi) to power -1, by the integral from minus infinity to plus infinity, by d omega exp , considered as a function of t dash, has some remarkable properties. The expression is known as the DIRAC delta-FUNCTION and is denoted by delta (t- t dash).

Quantitatively speaking, since the left-hand side of (8.46) is independent of the value of f (t dash) for all t dash not equal to t, and f itself is an arbitrary function, the delta-function must have the effect of making the integral over t dash receive zero contribution from all t dash except in the immediate neighbourhood of t dash = t, where the contribution is so large that a finite value for the integral results."

Source: "Mathematical Methods For The Physical Sciences. An Informal treatment for students of physics and engineering." by K.F. Riley (Lecturer in physics, Cavendish Laboratory. fellow of Clare College, cambridge) (copyright Cambridge University Press 1974) (pages 214, 215)

I assume copoyright law allows me to quote this maths book in discussion here.

While the math detail is out of my depth; I can see enough in their explanation to see this whole thing is Dick Stafford's paper through and through.

Consider:

"left-hand side refers to a value for f at a PARTICULAR value of t, whilst right-hand side contains an integral over ALL values of the argument of f..."

this looks like one perspective on the left that can be seen from many perspectives on the right

or narrowing on the left and broadening on the right

that is, a partial differential equation that defines "definition" itself?

"Now, since the left-hand side of (8.46) is independent of the value of f (t dash) for all t dash not equal to t and f itself is an arbitrary function..."

t dash looks rather like a specific subset of an examined set

"...the delta function must have the effect of making the integral over t dash receive zero contribution from all t dash except in the immediate neighbourhood of t dash = t, where the contribution is so large that a finite value for the integral results."

it looks like here t is the examined set of Dick's set of numbers. Only a t perspective on t dash allows a finite value for the integral over t dash? It sems to be about creating a finite structure, a perspective, from two other perspectives that map to each other.

"The delta function can be visualiuzed as a very sharp narrow pulse (in space, time, density, etc.) producing an integrated effect of definite magnitude."

(re" Kronecker delta: page 178: "delta little i little j, which has the value 1 when i = j and the value 0 when i not equal j, is known as the Kronecker delta.) It seems to me that the Kronecker delta is not what Dick's work is about, but I can see how it might look like it from a certain restricted perspective.

-dolphin

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