so i know what it is (i think lol) ...
but what are its applications?
Reimann mapping is useful in branches of circuit theory and functions on the complex plane (lambert W...ect).
What is most facinating about the field are the generation of Reimann spheres using sphere orgin and termination at 0 and infinity respectively.
the riemann maping theorem tells you there are exactly 3 simply connected riemann surfaces, the plane, the disc, and the sphere.
it is always useful to know all possible objects of any kind.
since every riemann surface is the image of a covering ampping from a simply conected one, this says there are three kinds of riemann surfaces altogetehr, those whose covering space is the sphere, the disc and the plane. among compact riemann surfaces, it turns out the sphere covers only the sphere, and the plane covers only the surfaces of genus one, and the disc covers all the rest.
this gives the basic division of the world of surfaces into three types, parabolic, hyperbolic and elliptic. i.e. curvature zero, negative, or positive.
here is an example of a very powerful theorem that is proved not just by knowing the riemann mapping theorem classifying all simply connected riemann surfaces, but knowing which one covers a certain set.
Fact: the simply connected covering space of the plane minus 2 points is the disc. hence if any entire function misses two points then it factors through a holomorphic map into the disc, which is constant by some standard theorem (any bounded entire function is constant, which follows from the cauchy integral formulas for the derivative)), hence so was the original function.
i.e. (picards little theorem) any entire function missing two values is constant.
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