point at infinity

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1: 1.13 Differential Equations

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Transformation of the Point at Infinity

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2: 23.20 Mathematical Applications

… ►The curve $C$ is made into an abelian group (Macdonald (1968, Chapter 5)) by defining the zero element $o=(0,1,0)$ as the point at infinity, the negative of $P=(x,y)$ by $-P=(x,-y)$, and generally $P_1+P_2+P_3=0$ on the curve iff the points $P_1$, $P_2$, $P_3$ are collinear. …

3: 1.10 Functions of a Complex Variable

… ►A function whose only singularities, other than the point at infinity, are poles is called a meromorphic function. If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles. …

4: 21.7 Riemann Surfaces

… ►Equation (21.7.1) determines a plane algebraic curve in ${ℂ}^2$, which is made compact by adding its points at infinity. …

5: 28.2 Definitions and Basic Properties

… ►This equation has regular singularities at 0 and 1, both with exponents 0 and $\frac{1}{2}$, and an irregular singular point at $\mathrm{\infty }$. …

6: 2.7 Differential Equations

… ►To include the point at infinity in the foregoing classification scheme, we transform it into the origin by replacing $z$ in (2.7.1) with $1/z$; see Olver (1997b, pp. 153–154). …

7: 1.9 Calculus of a Complex Variable

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§1.9(iv) Conformal Mapping

►The extended complex plane, $ℂ\cup \{\mathrm{\infty }\}$, consists of the points of the complex plane $ℂ$ together with an ideal point $\mathrm{\infty }$ called the point at infinity. …

8: 6.4 Analytic Continuation

… ►Analytic continuation of the principal value of $E_1\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\mathrm{\infty }$. …

9: 15.2 Definitions and Analytical Properties

… ►As a multivalued function of $z$, $𝐅(a,b;c;z)$ is analytic everywhere except for possible branch points at $z=0$, $1$, and $\mathrm{\infty }$. …

10: 16.2 Definition and Analytic Properties

… ►Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at $z=0,1$, and $\mathrm{\infty }$. …