I don’t understand this integral in an expression. It’s written as the definite integral of dsigma sigma^-2, with the limits of integration being 0 and infinity. That is, there’s an integral sign, then there’s “dsigma”, then there’s “sigma^-2”. I’m typing out “sigma” here but it’s the lowercase Greek in my source.
Pathetically, I read this as either of two possibilities:
the integral of nothing, which would be zero, multiplied by one over the square of sigma, all of which would be trivially zero;
or else as the integral of sigma^-2, but I don’t get why sigma^-2 is written outside of the integral if it’s to be integrated.
It’s from a published paper (1990 in Sweden) but I can’t find the paper for free online to point anybody to. The paper is “Transient plane source techniques for thermal conductivity and thermal diffusivity measurements of solid materials” by Silas E. Gustafsson. And, no, it’s not homework.
The I-sub-0 is a modified Bessel function of the first kind, for n=0. I assume the “X” just means multiplication and everything here is a scalar, and I expect D(tau) to have a scalar value, though the use of u and v and some physical considerations do make me slightly afraid the “X” refers to a cross product. At this point the way to read the integral I’m asking about is my biggest confusion.
While I was taught ∫<integrand(x)>dx originally in undergrad, I have seen many papers in physics and finance where the notation ∫dx<integrand(x)> is used to mean the same thing (with what is “inside” and “outside” the integral implied by context, rather than what is between the “∫” and the “dx”).
I suspect this means that your integral is really just a simple triple integral of u and v from 0 to 1, and sigma from 0 to tau.
Although I have seen people write the differential before the integrand, it is not usual and if \sigma^{-2} is indeed the integrand, then the integral blows up. Also the integrals of u du and of v dv between 0 and 1 are each 1/2 whether or not they inside the d\sigma integral or not. A very mysterious expression, all in all.
The differential element dx needn’t be at the end, and notationally and/or pedagogically there are many times when it’s convenient to put it elsewhere. In your example, perhaps the arrangement maintains a parallel structure with an upstream line? Also, for all three integrals in your example, the differential elements would go all the way to the very far right-hand end of the second line if you wanted them to be the “capstone” of the integrand, but that takes them far away from their respective integral signs and integration limits. Keeping them close helps one keep things straight. That “x” sign is just linking the two lines together.
Here is another way to write the expression, but notice that it is less transparent as to which integration range goes with which integration variable.
The notation means that there’s a factor of 1/sigma^2 in the integral; that’s all. Note that the integrand doesn’t blow up as sigma goes to zero except if u = v = 0; the limit of exp(-A/sigma^2) is zero as sigma goes to zero, faster than 1/sigma^2 factor blows up, unless A = 0.
Really? Jeez, I was hoping to evaluate this numerically to create a graph over tau. I think this means I’d have to step through tau, and through sigma, u, and v, and also I’d have to step through another variable to evaluate the Bessel function (the variable mathworld.wolfram.com refers to as “k”). That’s five nested levels of iteration. If I give each one just 100 steps, that’s ten billion evaluations, which are going to include a factorial and various power functions.
Is this an expression that’s just not practical to use in numerical software?
Ten billion doesn’t sound so bad. Even with the Bessel function, the time to produce the graph will be measured in minutes or hours, not months. With a little thought toward choosing optimal step sizes for each nested level, you should be able to get what you need through numerical integration.
