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Microlocal analysis, harmonic analysis, quantum ergodicity etc.

Nothing at usyd undergrad I don't think (although Lyapunov exponents were briefly mentioned in a 3rd year biomathematics course). It's pretty niche. If there are any academics in the department working in chaos (I can't think of any of the top of my head but there are many I haven't met), then...

There is a generally agreed upon definition for chaos in a dynamical system (a system which evolves in a discrete or continuous time variable). 1. It must be sensitive to initial conditions. (The butterfly effect.) This can be made rigorous using Lyapunov exponents, which measure how quickly...

Yep, I mostly went with things somewhat connected to my interests...as the time I have to devote to such a blog will be a bit limited. Some aspects of numerical analysis and computational mathematics interest me a bit, but relatively not as much as most of the things I listed. That said...

But this is exactly the thing you are trying to improve! Avoiding them because you aren't good at them is incredibly counterproductive. Attempt them until you get good at them.

=2f(x)\cdot\lim_{h\rightarrow 0}\frac{f(x+3h)-f(x-h)}{h} &=& 2f(x)\cdot (\lim_{h\rightarrow 0}\frac{f(x+3h)-f(x)}{h}+\lim_{h\rightarrow 0}\frac{f(x)-f(x-h)}{h}) &=&2f(x)\cdot(3f'(x)+f'(x))=8f(x)f'(x). Edit: muscles flexed.

Oh wow, okay yeah I see. Well its easy enough to understand the exp function graphically on the complex plane (its the familiar exponential on the real line and both real and complex part oscillate sinusoidally in the imaginary direction), so that is a start. The logarithm is easy enough too...

Kind of related, but probably not a satisfactory answer to your question: http://community.boredofstudies.org/238/extracurricular-topics/329588/what-does-mean-have-irrational-complex-powers.html. I don't know quite what you are hoping for in terms of a visualisation.

They are just leading order Taylor approximations.

Are you sure about that answer? I think (*) is incorrect, as well as the Maclaurin series. I think it should be y(x)=x-x^3/2 to 3rd order (and y'''(0)=-3).

Whenever you see two competing limiting behaviours you should think L'Hopital.

For the limit one, its just \lim_{n\rightarrow \infty} n(c^{1/n}-1)=\lim_{t\rightarrow 0^+} \frac{c^t-1}{t}=c^t\log(c)|_{t=0}=\log(c) by L'Hopital.

Re: MX2 Integration Marathon It's just convention, and what people have pretty much stuck to. By calling the functions I described as *elementary*, we aren't making any mathematical claims about them, we are just deciding on a name for a certain class of functions that come up a lot in low...

Re: MX2 Integration Marathon Its a pretty nice and simple sum lol.

Re: MX2 Integration Marathon Well the precise defn would be: CAN be obtained by a finite recursive application of the allowed operations on the listed basic functions. sin(x) can be obtained by applying absolutely 0 operations as it is itself one of the basic functions. Hence it is...

Re: MX2 Integration Marathon I'm pretty sure all constant functions are considered elementary, no matter how unusual the number. (Most numbers are transcendental anyway, which imo is weirder than being a root of a poly that cant be expressed by radicals). Also, the recursive building of an...

Re: MX2 Integration Marathon I would say something like Elementary function = anything formed recursively (using the operations of +,-,*,/,^ and composition) from: the reals, the monomial x, the 3 trig funcs and their inverses, the 3 hyp trig funcs and their inverses, the exponential and the...

a) The last 6 rolls must be 6's. The 7-th roll must be a 1-5 (else we would have an earlier termination) The first 6 rolls can be anything as long as they aren't 6 consecutive 6's. So P_a=(1/6)^6(1-(1/6)^6)(5/6). b) Take the complement, probability of termination on 7th,8th,9th rolls are...

Re: MX2 Integration Marathon I didn't forget, I just didn't bother. Any rational function can just be done by partial fractions, which isn't terribly exciting. I=\int \frac{1}{2(x^2-1)}+\frac{1}{4(x^2+x+1)}+\frac{1}{4(x^2-x+1)}\, dx =\frac{1}{4}\log(\frac{1-x}{1+x})+\frac{1}{2...

Re: MX2 Integration Marathon I do realise that this particular question can be done, seeing as I literally did it in the post you are quoting. If you think you have found an elementary primitive of e^{-x^2} on the other hand (which is what I am claiming is impossible when I say we can't...