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Apr
22
comment How does the Dirac delta function operate when its peak is at the boundary of an integral?
In retrospect, it's possible this could have been written in a way that would make it a good question for Physics. Such a question would ask about the physical reasoning for using a particular convention for the integral of a delta function, in some specific situation.
Apr
22
comment How does the Dirac delta function operate when its peak is at the boundary of an integral?
@AAM the example I'm familiar with is Feynman diagrams involving a gluon emitted with a fraction $1-\xi$ of the forward momentum of the parent particle. We integrate over $\xi$ to take into account all possible gluon momenta, and $\delta(1-\xi)$ represents the contribution of a diagram with no emitted gluon. In that case $\xi > 1$ is physically meaningless; it wouldn't make sense to consider our deltas as the limit of symmetric peaked functions, but they could be the limit of asymmetric functions (defined only on $\xi < 1$), for which $\int^1 f(\xi)d\xi = 1$.
Apr
13
comment How does the Dirac delta function operate when its peak is at the boundary of an integral?
@AccidentalFourierTransform indeed, I routinely work with situations where the correct result is $\int_a^b \delta(x-a)f(x)dx = f(a)$. I suppose this means an ideal answer would address the different possible conventions and any factors that might affect which one a person should choose.
Apr
8
comment Asymptotic behavior of an integral involving the gamma function
Thanks; I'm not clear on a couple things, though. First, going from (2) to (1) do I understand correctly that you use De l'Hopital's thm with numerator $\int_0^1(\ldots) d\xi$ and denominator $1/\sqrt{k}$? If so, how do we know a priori that the large-$k$ limit of the numerator is zero? Also, I don't follow how you went from $f_k(\xi)$ to $\widetilde{f}_k(\xi)$. I'm only familiar with stationary phase in diff eqs, and I don't see how De l'Hopital supports that step either. Could you elaborate or link me to a source for further reading?
Apr
7
comment Asymptotic behavior of an integral involving the gamma function
@user1952009 No, I didn't think of that. Would you care to put it as an answer?
Apr
7
asked Asymptotic behavior of an integral involving the gamma function
Apr
4
comment Evaluate the sum with special function
In the very first part of your display equation, did you mean to use $j$ instead of $n$ in the expression?
Apr
1
awarded  Popular Question
Mar
29
comment Why is cardinality of set of even numbers = set of whole numbers?
@ThomasW I suggest looking up measure theory, which is one area where concepts like density become relevant.
Mar
24
comment Why are turns not used as the default angle measure?
@Semiclassical This would be in a context where a few orders of magnitude lost here or there are not a problem. Like estimating how many stars are in the universe, or the field strength of a graviton (where the real answer is "way too weak to ever detect" and a few orders of magnitude won't change that).
Mar
21
answered How to prove $\exp(x)/(\exp(x)+1)^2$ is even?
Mar
16
awarded  Popular Question
Mar
13
comment Is it possible to prove uniqueness without using proof by contradiction?
@Voyska I think the confusion was that $\emptyset$, $1$, and $0$ are not proofs. You say "...several proofs of uniqueness:" and then give a list, and one would expect that the items in the list are proofs of uniqueness. What you've actually shown are objects that satisfy uniqueness theorems. If you want to improve the question, you should probably state the theorems, e.g. that $\emptyset$ is the only empty set, that $1$ is the only multiplicative identity, etc.
Mar
12
comment How to tell what dimension an object is?
@frog1944 if you are able to represent a vector in a form with rows, you've already found the dimension of the space. A vector isn't really a set of numbers. That's only a representation of a vector. The vector itself is an abstract mathematical object that can be added to other vectors and multiplied by numbers to produce other abstract mathematical objects.
Mar
12
comment How to tell what dimension an object is?
A bit of feedback, if I may: the question seems to be getting at some kind of procedure that you can apply to a set of points (though obviously not an arbitrary set) to produce the set's dimension. This answer talks about representing points in, say, $(0,1)^2$ as a vector space which turns out to have two dimensions. But one might then wonder why you can't do the same thing with a space-filling curve, thereby representing $(0,1)^2$ as a one-dimensional vector space. It feels a little incomplete not to address that. (Granted, that might be more than an answer's worth.)
Mar
12
comment determinant of a very large matrix in MATLAB
For clarity it might help to put the formula for the determinant itself in the post ($\sqrt{n^n}$), or italicize the "log", or some such thing. I think this is an easy reading mistake to make.
Mar
8
comment Recovering a quadratic polynomial from three values using calculus
@user109256 If there were something wrong with it, people would probably downvote, and maybe comment to let you know what they think is wrong. If you just haven't gotten upvotes, it probably means that not many people have seen the question, and those who have seen it didn't think it was especially useful. That's quite common. (Of course, now you do have some upvotes.)
Feb
26
comment How does law of cosines work with vectors?
This is a good question, but it seems not to have a physical context. Perhaps we should send it to Mathematics?
Feb
22
comment How to prove $b=c$ if $ab=ac$ (cancellation law in groups)?
@user1717828 I don't think this has anything to do with being a physicist - I've always thought it was a symptom of bad (or insufficiently deep) math education. It's very common among students of all quantitative fields that they memorize the rule of cancellation without understanding why things can be cancelled, and that gets them into trouble when dealing with more sophisticated situations like matrices, modular arithmetic, differential operators, etc.
Feb
21
comment Converting decimal fractions to base N
@SumMathGuy it's the same thing going on, you just go from smallest to largest power rather than largest to smallest. This happens to work because there are no negative powers in the number (i.e. it's an integer). For a number with no positive powers (a decimal between 0 and 1), you can do something similar, but multiply instead of divide: $0.5\times 3 = 1.5 = 0.5\text{ r } 1$, so $1$ is the first digit, and so on. As a bonus, it's really obvious this way that the decimal repeats. However, for a general non-integer number you have to do the integral and fractional parts separately.