# How to find the second derivitive of delta function?

• KFC
In summary, the conversation discusses finding the second derivative of a delta function and how it applies to a complex parameter. It is not possible to formulate a consistent general theory for this, but the book mentioned may provide some insight.
KFC
how to find the second derivitive of delta function?

let f=the dirac delta
D=differentiation
[D^n]f=[n!(-1/x)^n]f
or
{[D^n]-[n!(-1/x)^n]}f=0
if n=2
[D^2]f=[n!(-1/x)^2]f

Thanks. It helps. What about if x is complex?

KFC said:
Thanks. It helps. What about if x is complex?

It is not possible to formulate a consistent general theory for that. What specifically did you want to do?

I want to know how to carry out first derivitative of a delta function with parameter is complex. For example, $$\delta(z)$$, where $$z=x+iy$$

how to carray out $$\frac{d\delta(z)}{dz^*}$$ where $$z^*$$ mean the conjugate of z

The definition of the derivative of the delta function is

$$\int dx~\delta^{(n)}(x-y)f(x) = (-1)^n f^{(n)}(y)$$

Formally, you integrate by parts to get

$$\int dx \delta^{(n)}(x-y)f(x) = (-1)^{n}\int dx \delta(x-y) f^{(n)}(x)$$.

If it's at all possible to generalize this to the case you want, you should start with a method like this. I haven't run into a complex delta function, though, so I don't know if such as thing is well defined, let alone it's derivative (and especially its derivative with respect to the complex congugate).

This book seems to discuss the complex delta function:

Last edited:

## 1. What is a delta function?

A delta function, also known as the Dirac delta function, is a mathematical tool used in calculus and physics to represent a point-like singularity. It is defined as zero everywhere except at the origin, where it is infinite, and its integral over any interval containing the origin is equal to one.

## 2. What is the role of the second derivative of a delta function?

The second derivative of a delta function is used to describe the curvature or sharpness at the origin. It can also be used to represent the rate of change of the first derivative of a function at the origin.

## 3. How do you find the second derivative of a delta function?

To find the second derivative of a delta function, you can use the definition of the derivative and apply it to the first derivative of the delta function. This will result in a double derivative of the delta function, which can then be evaluated at the origin to find the second derivative.

## 4. What is the difference between the first and second derivative of a delta function?

The first derivative of a delta function represents the slope or rate of change at the origin, while the second derivative represents the curvature or sharpness at the origin. Essentially, the first derivative describes the behavior of the delta function near the origin, while the second derivative describes the behavior of the first derivative.

## 5. Why is the second derivative of a delta function important in physics?

In physics, the second derivative of a delta function is used to describe the behavior of particles or systems near a point-like singularity. It is commonly used in quantum mechanics to represent the location of a particle, and its second derivative can provide information about the particle's energy and momentum.

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