In one attempted proof of this theorem, William Osgood took issue with the implications when the boundary is not a simply closed Jordan curve; in that attempt, Axel Harnack used what is now Harnack's theorem and established the existence of a Green's function.
In one attempted proof of this theorem, William Osgood took issue with the implications when the boundary is not a simply closed Jordan curve; in that attempt, Axel Harnack used what is now Harnack's theorem and established the existence of a Green's function. One method of proving this theorem involves taking the normal family of analytic functions that send a fixed z-naught to zero and finding a member function with the supremum value for f prime of z-naught; modern methods of proving it often involve using a form of the Schwarz lemma. The namesake of this theorem coined the name for (and arguably relied on) the result from potential theory now called Dirichlet's principle(*), which can be used to complete the existence proof for this theorem. The uniformization theorem can be seen as the same result for surfaces as this theorem gives for plane regions. For 10 points, name this theorem first described in its formulator's 1851 PhD thesis, which shows the existence of a bijective holomorphic function which takes a non-empty simply connected open subset of the complex numbers onto the open unit disk.
In one attempted proof of this theorem, William Osgood took issue with the implications when the boundary is not a simply closed Jordan curve; in that attempt, Axel Harnack used what is now Harnack's theorem and established the existence of a Green's function.
Riemann mapping theorem