Principal value of an integral

[chiraganand]

Can anyone tell me what is the principal value of an integral? The integral in this case is a surface integral?

 

[RUber]

The principal value of an integral is normally thought of as $\lim_{n \to \infty } \int_{-n}^n f(x) dx$. It is a way of dealing with infinite integrals which have some symmetry.
Another application uses the limit approaching a point of discontinuity...like if there is a break at x = c on an interval [a,b], you could have a principal value integral of $\int_a^b f(x) dx \equiv \lim_{\varepsilon \to 0 } \left( \int_a^{c-\varepsilon} f(x) dx + \int_{c+\varepsilon}^b f(x) dx \right)$

 

[chiraganand]

The principal value of an integral is normally thought of as $\lim_{n \to \infty } \int_{-n}^n f(x) dx$. It is a way of dealing with infinite integrals which have some symmetry.
Another application uses the limit approaching a point of discontinuity...like if there is a break at x = c on an interval [a,b], you could have a principal value integral of $\int_a^b f(x) dx \equiv \lim_{\varepsilon \to 0 } \left( \int_a^{c-\varepsilon} f(x) dx + \int_{c+\varepsilon}^b f(x) dx \right)$

So how do i know if an integral is a principal value integral?

 

[RUber]

One indication would be having it written as a limit.
If you are asking if you may use a principal value method to evaluate an integral, then that depends on your application. What specific surface integral are you working with?

 

[chiraganand]

One indication would be having it written as a limit.
If you are asking if you may use a principal value method to evaluate an integral, then that depends on your application. What specific surface integral are you working with?

Actually i am looking at the integral representation of scattering problems using sommerfeld radiation conditions. So i mean scattering of a wave from a flaw in a fluid. I have attached the pages of the book i am looking at . The first pic is of the sommerfeld radiation conditions and in the next page we have integral equations for scattering problems in which it says that p.v. dentoes the principal value integral

 

[RUber]

Got it. This looks like you are using the p.v. to avoid the discontinuity. See it as the limit as distance from the surface goes to zero for those normal components.

 

[chiraganand]

Got it. This looks like you are using the p.v. to avoid the discontinuity. See it as the limit as distance from the surface goes to zero for those normal components.

sorry for acting really dumb.. but what does that mean??

 

[RUber]

It probably depends on your Green's function. Notice that when $\frac{\partial G}{\partial n}$ is in the integral, you have to use P.V. Many of these Green's functions have hyper-singular derivatives which cannot be simply removed.
In your case, $G(R) =\frac{e^{ikR}}{4\pi R}$ is singular for R = 0, but still a function that can be integrated.
$\frac{\partial G}{\partial n}(R) =\frac{ik e^{ikR}}{4\pi R} -\frac{ e^{ikR}}{4\pi R^2}$ Which has one higher degree of singularity at R = 0. This fact precludes it from traditional integration. Thus you are using the principal value, essentially removing the point R=0 from your integration.
Also look at Hadamard finite part integration. These seem to be about the same.

 

[chiraganand]

It probably depends on your Green's function. Notice that when $\frac{\partial G}{\partial n}$ is in the integral, you have to use P.V. Many of these Green's functions have hyper-singular derivatives which cannot be simply removed.
In your case, $G(R) =\frac{e^{ikR}}{4\pi R}$ is singular for R = 0, but still a function that can be integrated.
$\frac{\partial G}{\partial n}(R) =\frac{ik e^{ikR}}{4\pi R} -\frac{ e^{ikR}}{4\pi R^2}$ Which has one higher degree of singularity at R = 0. This fact precludes it from traditional integration. Thus you are using the principal value, essentially removing the point R=0 from your integration.
Also look at Hadamard finite part integration. These seem to be about the same.

Thanks for the info and the help.. any suggestions on books which i can refer to for all these functions? I am being bombarded with a lot of functions throughout this course for ex hankel bessel green etc etc

 

[RUber]

There are lots of books out there on special functions. But they can be terrible to read. I recommend taking it one bite at a time. As you learn one function, ask what it is doing at zero and infinity, and where the zeros of the function itself are. You can say a lot about the function with just that information.

 

[chiraganand]

There are lots of books out there on special functions. But they can be terrible to read. I recommend taking it one bite at a time. As you learn one function, ask what it is doing at zero and infinity, and where the zeros of the function itself are. You can say a lot about the function with just that information.

ok thanks! am going to be troubling you guys a lot.. facing a lot of problems in the literature.. lot of mathematical functions.. thanks a ton though!