Re: [math-fun] Jacobian and related
Now assume further that the entire function f : C —> C is onto. (Unlike exp(z), which omits 0.) Fun question: ----- Does there exist an analytic function f : C —> C that is a) locally one-to-one (equivalently, f'(z) is nowhere zero), and b) onto ??? ----- —Dan Gene Salamin wrote: ----- On Saturday, June 8, 2019, 11:12:30 PM PDT, Dan Asimov <dasimov@earthlink.net> wrote: ----- Obviously a *necessary* condition is that P be locally one-to-one Q x Q —> Q. Which is just saying that for all (x,y) in Q^2, we have dP/dx * dP/dY ≠ 0 (i.e., not both derivatives are zero). ------------------------------------------------------------ The relation dP/dx * dP/dY ≠ 0 does not imply that P is locally one-to-one.. Counterexample: P(x,y) = xy. ---------------------------------------------------------------- Puzzle (tricky!): Find an entire function f(z) — i.e., f(z) is analytic on all of C — that is locally one-to-one everywhere without being globally one-to-one ... or else prove that such cannot exist. —Dan -------------------------------------------------————— f(z) = exp(z). --------------------------------------------------------------------- -----
The only analytic bijections of the Riemann sphere are the homographic functions w=(az+b)/(cz+d). If the function is analytic on the finite complex plane, it cannot have a pole, so c=0. Thus the only such functions are linear, w=az+b. -- Gene On Sunday, June 9, 2019, 10:52:44 AM PDT, Dan Asimov <dasimov@earthlink.net> wrote: Now assume further that the entire function f : C —> C is onto. (Unlike exp(z), which omits 0.) Fun question: ----- Does there exist an analytic function f : C —> C that is a) locally one-to-one (equivalently, f'(z) is nowhere zero), and b) onto ??? ----- —Dan Gene Salamin wrote: ----- On Saturday, June 8, 2019, 11:12:30 PM PDT, Dan Asimov <dasimov@earthlink.net> wrote: ----- Obviously a *necessary* condition is that P be locally one-to-one Q x Q —> Q. Which is just saying that for all (x,y) in Q^2, we have dP/dx * dP/dY ≠ 0 (i.e., not both derivatives are zero). ------------------------------------------------------------ The relation dP/dx * dP/dY ≠ 0 does not imply that P is locally one-to-one.. Counterexample: P(x,y) = xy. ---------------------------------------------------------------- Puzzle (tricky!): Find an entire function f(z) — i.e., f(z) is analytic on all of C — that is locally one-to-one everywhere without being globally one-to-one ... or else prove that such cannot exist. —Dan -------------------------------------------------————— f(z) = exp(z). --------------------------------------------------------------------- -----
On Sun, Jun 9, 2019, 14:08 Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
The only analytic bijections of the Riemann sphere are the homographic functions w=(az+b)/(cz+d). If the function is analytic on the finite complex plane, it cannot have a pole, so c=0. Thus the only such functions are linear, w=az+b.
This shows impossibility under the added assumption that the function can be extended from the complex plane to the Riemann sphere, but that additional assumption is not needed. F is a covering map on the plane. Since the plane is simply connected, it can't have more than one sheet. Andy
-- Gene
On Sunday, June 9, 2019, 10:52:44 AM PDT, Dan Asimov < dasimov@earthlink.net> wrote:
Now assume further that the entire function f : C —> C is onto. (Unlike exp(z), which omits 0.)
Fun question: ----- Does there exist an analytic function f : C —> C that is a) locally one-to-one (equivalently, f'(z) is nowhere zero), and b) onto ??? -----
—Dan
Gene Salamin wrote: ----- On Saturday, June 8, 2019, 11:12:30 PM PDT, Dan Asimov < dasimov@earthlink.net> wrote: -----
Obviously a *necessary* condition is that P be locally one-to-one Q x Q —> Q.
Which is just saying that for all (x,y) in Q^2, we have
dP/dx * dP/dY ≠ 0
(i.e., not both derivatives are zero).
------------------------------------------------------------
The relation dP/dx * dP/dY ≠ 0 does not imply that P is locally one-to-one.. Counterexample: P(x,y) = xy.
----------------------------------------------------------------
Puzzle (tricky!): Find an entire function f(z) — i.e., f(z) is analytic on all of C — that is locally one-to-one everywhere without being globally one-to-one ... or else prove that such cannot exist.
—Dan -------------------------------------------------—————
f(z) = exp(z).
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participants (3)
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Andy Latto -
Dan Asimov -
Eugene Salamin