# Interesting integral property

• CPL.Luke
In summary, the problem the person was having was trying to solve the schroedinger equation in the presence of a potential, and they were trying to do it through the use of a Fourier transform. They were not able to understand what d^2 u meant inside the integral, so they deduced that it must exist from integration by parts. However, I doubt that this property actually holds for any function g(x) and u(x).

#### CPL.Luke

so I'm having trouble with an integral. However I noticed an interesting property to it.int{gdu}=c*int{gd^2u}

with the integral being from -infinity to positive infinity.

The problem arose while I was trying to solve the schroedinger equation in the presence of a potential, I'm trying to solve it through the use of a Fourier transform, but in order to do that I need to know the Fourier transform of the potential.

can anybody lend a hand?

I don't understand what you mean by "d^2 u" inside the integral.

I found the property from integration by parts, the d^2u represents the differential of du.techincally the c came out of the g function when I integrated it, however I don't know whether or not it can be used to develop other properties of the integral.

if it helps at all g is equal to e^ikx and u is an unknown function which is the product of some known potential, and an unknown function psi.

The differential of du is zero. Did you mean something like u''(x) dx?

yes exactly

I'd be interested in knowing HOW you got that.

You are saying
$$\int_{-\infty}^\infty g(x)u'(x)dx= c \int_{-\infty}^\infty g(x) u"(x)dx$$
for any g(x) and u(x)? I doubt that is true. In particular, if you take u= x, that turns into
$$\int_{-\infty}^\infty g(x)dx= 0$$

no, as I said it was a property of a specific integral I was doing, where u is an unknown function, and g is equal to e^ikx, however running it through integration by parts I was able to deduce that that property exhisted, (because my function u goes to zero at -infinity and positive infinity, which is known because of other known properies of u)

the integral came up as I was experimenting with Fourier transforms of the schroedinger equation, I'm hoping that there may be some trick that would simplify the problem.

## 1. What is an "interesting integral property"?

An interesting integral property refers to a mathematical property of an integral that is unique, unexpected, or has practical applications. It can also refer to a property that makes an integral easier to solve or evaluate.

## 2. How do you determine if an integral property is interesting?

Determining if an integral property is interesting is subjective and can vary depending on the context. Some may consider a property interesting if it has practical applications or simplifies the integral, while others may find a property interesting if it is unexpected or unique.

## 3. Can you provide an example of an interesting integral property?

One example of an interesting integral property is the substitution rule, which states that if u = g(x) is a differentiable function, then the integral of f(g(x))g'(x)dx can be rewritten as the integral of f(u)du. This property allows for easier evaluation of integrals by substitution of variables.

## 4. Are interesting integral properties important in scientific research?

Yes, interesting integral properties are important in scientific research as they can provide insights and simplify calculations in many fields, such as physics, engineering, and economics. They can also lead to new discoveries and techniques in solving complex problems.