so g(x) is another function we wish to find, with the property that f(x)'s level curves intersect g(x)'s level curves at right angles (wherever they do intersect)
for example, let f(x,y)=x^2+y^2
the level curves are of the form x^2+y^2=A, i.e. circles
aha! so there is context behind your questions. you have somehow been assured that you need to learn about orthogonal trajectories in the context of complex analysis because that topic may be on a test
but you will need to be more specific about what you mean about "the case of complex functions," since they map the plane to other points in the plane, rather than the plane to real values (so you cannot define level curves in the same sense). it is true that the real and imaginary parts will each be real-valued functions of the plane whose level sets form orthogonal trajectories, though
I have no idea what the uni means by orthogonal trajectories. They just put it under the chapter of complex analysis. What do u think I should learn ? Your best judgement ?
really, all you need to know is the "case" with real-valued functions, and how that relates to complex variables (the relation that I just mentioned, real and imaginary parts)
yes, a function f:U->C (where U is an open subset of C) is complex-analytic at z (z in C) iff f's partial derivatives at z satisfy the cauchy-riemann equations
"if I plug in a from A then I should get some element from B" describes any function f:A->B before even beginning to talk about level sets (aka level curves, fibers, many other names)
@amWhy lol I waste so much printer paper, I usually just erase my work and write over it instead of getting new paper, there are eraser crumbs all over my desk
hahahaha. I understand. I go through erasers as quickly as I go through pencils!! No, I don't use slate, though I really should think about a portable black-board (or whiteboard)...but I like my "fine point" pencils.
@Ethan Basic complex analysis, some number theory, physics olympiad problems and x-ray crystallography (which means Fourier transforms later). Right now I'm doing number theory. You? A
@Ethan The Crystallography's for a school project, and there's (from my still fairly ignorant viewpoint) a lot of overlap between (analytic) number theory and complex analysis
Although really, I'm just sampling different areas of maths and physics and sticking with what seems interesting. What are you studying?
@Alyosha sorry, was busy im pretty much just studying number theory and bits of analytic number theory, hopefully I will have time to study some enumerative combinatorics before my break is over, I have only been studying mathematics really on my own time for about 2 or 3 years now.
i am not an expert on the topic, but I think sometimes there is ambiguity when one has a function represented by some expression valid for one interval, and then sees it being expressed in this matter for things not in that interval
and there are probably a variety of other uses, I don't know to much complex analysis ( or analysis in general ) so I don't typically deal with this sort of thing
@nick en.wikipedia.org/wiki/Zeta_function_regularization, "An example of zeta-function regularization is the calculation of the vacuum expectation value of the energy of a particle field in quantum field theory. More generally, the zeta-function approach can be used to regularize the whole energy-momentum tensor in curved spacetime."
has some more stuff about physics, summation methods, and the zeta function in particular
Though I barely know some basic mechanics, so I can't really say anything about the physics stuff lol
ye any multiplictive function so that the domain of $f$ is the integers and its range is the complex numbers
anyway well it turns out there are some very nice identities that turn out from the theory of modular forms, like $$\sum_{k=1}^{n-1}\sigma_3(k)\sigma_3(n-k)=\frac{\sigma_7(n)-\sigma_3(n)}{120}$$
@Alyosha this also ventures into the theory of elliptic functions which is very beautiful, and which I know nothing about, things like $$\sum_{n=1}^\infty\frac{1}{e^{\pi n^2}}=\frac{\pi^{1/4}}{2\Gamma(3/4)}-\frac{1}{2}$$ amaze me
@Alyosha the series is related to heegner numbers, which are related to several mathematical 'coincidences' like the almost integer, $$e^{\pi\sqrt{163}}=262537412640768743.999999999999..$$
@Alyosha this sort of thing eisenstein series/ elliptic functions etc are definitely topics I want to study, though I am lacking in complex analysis and a variety of other fields commonly used in there study
Well the fundamental theorem of arithmetic says any integer $n$, can be written in a unique way as a product of prime powers $$n=p_1^{a_1}p_2^{a_2}p_3^{a_3}...=\prod_{k=1}^\infty p_k^{a_k}$$, where only a finite number of the $a_i$ are non zero, and for $n=1$, we can take them all to be zero
If two integers $n$ have the same factorization they are the same, and every integer has a factorization like this
And this sum runs over all the distinct tuples of exponents in the product of prime powers where each is a positive integer
but every integer corresponds to one of these tuples, because every integer can be written as a product of primes, with exponents being positive integers
For example $$\sum_{n=1}^\infty \frac{d(n)}{n^s}=\zeta(s)^2$$
@Alyosha theres a certain type of inverse integral transform, that involves complex analysis, and can be used to gives estimates on $$\sum_{n\leq x} f(n)$$ From $$\sum_{n=1}^\infty \frac{f(n)}{n^s}$$
@Alyosha google.com/… is the transform, basicly if $$F(s)=\sum_{n=1}^\infty\frac{f(n)}{n^s}$$ then $$\sum_{n\leq x}^* f(n)=\frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} F(z)\frac{x^z}{z} \ dz$$
The star on the sum indicates that the last term of the sum must be multiplied by 1/2 when x is an integer.
If x is not an integer it works fine, and gives the normal $$\sum_{n\leq x} f(n)$$
@Alyosha the formula requires that $F(s)$ be absoloutely convergent for $\Re(s)>c$
but other then that you can chose whatever c you want
@Alyosha these techniques were used in the first proof of the prime number theorem, and are still typically used for giving proofs of the prime number theorem
@Alyosha basicly in the proof of the prime number theorem, using this transform you can give an exact formula for the number of primes less then or equal to x, and a variety of other interesting functions, in terms of the zeros of the zeta function, the good thing is that the main term in most of these is a real well behaved expression like for $\pi(x)$ its $\text{Li}(x)$ the logarithmic integral
@Alyosha In the starred expression on the right substitute $$f(n)=\frac{\sigma_x(n)}{n^s}$$
And notice that for prime powers $p^j$
$$\sigma_x(p^j)=\frac{p^{x(j+1)}-1}{p^x-1}$$
then sum up the factors in the euler product using the geometric series formula, and simplify with a little bit of algebra
note that $$\zeta(s)=\prod_{p}\frac{p^s}{p^s-1}$$
so, $$\zeta(s)\zeta(s-x)=\prod_{p}\frac{p^{2s-x}}{(p^s-1)(p^{s-x}-1)}$$
@Alyosha one of arguably the most famous problems in mathematics is the reimann hypothesis which says the real part of all the non trivial zeros of the reimann zeta function will always have a value of $\frac{1}{2}$, if true this would give much better estimates then the best known now for $\pi(x)$ and other prime counting functions, it would also solve a bunch of other open problems in number theory and I think other fields to
though this is again getting pretty out of my scope of knowledge
When I say non trivial zeros, I am referring to zeros that have an imaginary part not equal to zero.
@Alyosha i am not sure I don't know much about the zeros of complex functions, this would be something I imagine covered in a text on complex analysis, I know however that there are many intuitive arguments for why it should be $\frac{1}{2}$ , and equivilent statements to the rh (short for reimann hypothesis)
@Alyosha here en.wikipedia.org/wiki/Riemann_hypothesis, on the same page if you scroll down it shows the explict formula for a varient of the prime counting function, and gives information on how its derived
Along with Reimann's century and a half old original paper written in german I think
you will learn about it when you study arithmetic functions (if you do lol)
its a useful arithmetic function
Though as I said again practically all of this is way out of my reach, as I have only been studying for a very short while, and haven't ever taken a course on or read anything thorough on analysis.
@Alyosha not really, but I remember reading somewhere that the summatory function for the mobius function (the mertens function its called), behaves like a random walk or somthing about it behaves that way, whatever that means lol
@Alyosha Reimann steiljes integrals are used in number theory sometimes to calculate sums
