The strange meaning of complex delta

[lokofer]

The strange meaning of .."complex delta"..

Let's suppose we introduce the function..

[tex] \delta (a+ix)=f(x) [/tex] a a real number.. then following the representation..

[tex] \frac{sin (Nx)}{x} [/tex] f(x) is oo elsewhere...except for pure complex numbers.

[tex] f(x)(2\pi) =\int_{-\infty}^{\infty}du e^{iua}exp(ux) [/tex]

f(x) is the "laplace inverse" transform of cos(as)

My question is if we can work or manipulate such function f(x) defined in the post although it makes no or little sense even considering it a "distribution"...

 

[lokofer]

- Oh by the way if you have the usual delta function [tex] \delta (x) [/tex]:

a) can you define the "composite" function [tex]F[ \delta (x)] [/tex] where F is a Real function

b) Is there an "F" so [tex] F[\delta (x)]= x [/tex]?

c) Can you define the [tex] D^{a} \delta (x) [/tex] where a is a Real or complex number ("Fractional calculus" with delta functions )

d) If you had an analytic function F so for every x or z( complex):

[tex] F(z)= \sum_{n=0}^{\infty} \frac{a(n)}{n!}z^{n} [/tex]

Could we make [tex] F[\delta (x)]= \sum_{n=0}^{\infty} \frac{a(n)}{n!}[\delta(x)]^{n} [/tex] ?

 

[tacman]

It seems like the content is discussing the complex delta function, which is commonly used in mathematics and physics to describe certain phenomena. The function has a strange meaning because it involves both real and imaginary numbers, which can be difficult to conceptualize or understand. However, it has important applications and can be manipulated and used in various mathematical operations. The use of the term "strange" may be referring to the abstract nature of the function and its unconventional properties. Overall, the complex delta function may seem strange at first, but it serves a purpose in mathematical and scientific contexts.