well changing the subject to the positive real axis for a minute, doesn't z^2 do this? for t small and positive, it sends e^(it) (t small, positive) to e^(2it). and e^(-it) to e^(-2it). of course if you have a magnitude r in there, it does change that into r^2, but you can't have everything.
so I should rephrase: first quadrant complex numbers get rotated towards the real axis but do not cross it, and the second quadrant is a reflection of the first, i.e., complex numbers get rotated towards the real negative axis but do not cross it
there oughta be a book called "1001 conformal mappings" that just delivers on the title. i have seen a few, mostly focused on fluid flow, that only give me like 100 conformal mappings.
i'm just afraid if you go using that as a title you will mainly get pure math books, which will not be the gallery of examples that you want to browse. i forget what the jargon for this was. i found a good book once in the engineering library, it was just all sorts of regions in the plane being mapped into other regions in various ways. it was almost a picture book (with formulas). 15 years ago sadly so i do not remember the title.
my guess is that an applied or even engineering book would be more useful for this goal than anything with 'complex analysis' as its title, although i could be wrong. i have seen complex analysis books spend a lot of time on conformal mapping.
Green's theorem. Let $P(x,y)$ and $Q(x,y)$ be continuous with continuous partials in a simply connected closed region $R$ whose boundary is the contour $C$. Then
$$\int_C Pdx+Qdy = \int\int_R (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy$$ where$C$ is traversed in the positiv...
https://ncatlab.org/nlab/show/topos#ElementaryTopos @Thorgott @copper.hat the nlab says toposes
this settles it
Notice as a point of orthography that 'topos' is a French word, formed from 'topologie.' and not a Greek word. In writing, Grothendieck always forms the plural according to the French rule for words ending in 's,' so it is invariant—'les topos.' So the English plural ought to follow the English rule—'toposes.'
Freyd. a confessed lover of classical endings and the inventor of cosmoi and logoi among other types of categories, says he heard that Grothendieck spoke of 'topoi' in Buffalo. I regard this as biased hearsay which can not stand against the published record.
@user2103480 The other uses of topos (literary, etc) in English, mainly use topoi as the plural (though, toposes is listed as an alternate). The mathematical might as well follow suit. Perhaps it would be best to accept topoi, as others should accept toposes, as a valid alternate. Perhaps there might be a finer distinction, sort of like cacti (many cactus plants of the same species) vs cactuses (many cactus plants of varying species), at play here.
@geocalc33 so f could presumably map an element of C to any complex number. g isn't going to do that. beyond that, not much to say. you could still regard g as a function from C to C if you wanted to. one is more than free to expand the codomain of a given function, if that is something one wants to do.
if you want to shrink the codomain of a given function, you can also do that, provided you don't shrink it so much that you start throwing out things in the range of the function.
@TedShifrin how much does being active in research matters for a PhD advisor? Would you say its better to look for advisors who are publishing more frequently? (or would you say this shouldn't be a criterion at all?)
so wait... for the group integers modulo 6, <2> = {0, 2, 4} is simultaneously a subgroup and a coset, but 1 + <2> = {1, 3, 5} is only a coset, right? Since it doesn't contain either an inverse nor an identity element.
@feynhat if you want a postdoc and academic research position ultimately, the reputation of the adviser and other recommenders is of paramount importance.
oh i'm sorry. like a true american i sometimes forget not everyone is in the US. if you are not in the US, people in the US often play jokes on april 1st. not everybody, or even most people, but some people.
but as ted says, probably not grad school admissions.
what you have up there was a joke about april 1 that fell flat. that might be sadder than an actual april fools joke.
got a really bonehead question here, sorry...trying to plug the following transfer function into some optimization software by converting to a system of ODEs:
where all the a's and 'b's are constants
and the transfer function is H(s) = R [another constant] * F(s) * U(s)
