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Given a holomorphic function f on the blue compact set and a point in each of the holes, one can approximate f as well as desired by rational functions having poles only at those three points. In complex analysis , Runge's theorem (also known as Runge's approximation theorem ) is named after the German mathematician Carl Runge who first proved ...
The non-calculator section is worth 43.75% of the exam score, while the calculator section is worth 18.75%. [5] Section II of the Exam includes 4 free response questions, with 2 not allowing a calculator and 2 allowing use of a calculator. Section II is worth 37.5% of the exam score, with the non-calculator and calculator sections weighed ...
For a torus, the first Betti number is b 1 = 2 , which can be intuitively thought of as the number of circular "holes" Informally, the kth Betti number refers to the number of k-dimensional holes on a topological surface. A "k-dimensional hole" is a k-dimensional cycle that is not a boundary of a (k+1)-dimensional object.
Rational functions can be either finite or infinite for finite values, or finite or infinite for infinite x values. Thus, rational functions can easily be incorporated into a rational function model. Rational function models can often be used to model complicated structure with a fairly low degree in both the numerator and denominator.
Every Laurent polynomial can be written as a rational function while the converse is not necessarily true, i.e., the ring of Laurent polynomials is a subring of the rational functions. The rational function f ( x ) = x x {\displaystyle f(x)={\tfrac {x}{x}}} is equal to 1 for all x except 0, where there is a removable singularity .
In it, geometrical shapes can be made, as well as expressions from the normal graphing calculator, with extra features. [8] In September 2023, Desmos released a beta for a 3D calculator, which added features on top of the 2D calculator, including cross products, partial derivatives and double-variable parametric equations.
The TI-81 was the first graphing calculator made by Texas Instruments.It was designed in 1990 for use in algebra and precalculus courses. Since its release, it has been superseded by a series of newer calculators: the TI-85, TI-82, TI-83, TI-86, TI-83 Plus, TI-83 Plus Silver Edition, TI-84 Plus, TI-84 Plus Silver Edition, TI-84 Plus C Silver Edition, TI-Nspire, TI-Nspire CAS, TI-84 Plus CE ...
Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing. The non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps (i.e. a non-orientable surface of (non-orientable) genus n). (This number is also called the demigenus.)