$#C+1 = 8 : ~/encyclopedia/old_files/data/A012/A.0102370 Analytic plane,
''complex-analytic plane''
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...given straight line in that plane. The coordinates of the points of a half-plane satisfy an inequality $ Ax + By + C > 0 $,
...plane, the latter is said to be closed. Special half-planes on the complex plane $ z = x + iy $
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...complex $z$-plane onto a neighbourhood of a point $w_0$ of the complex $w$-plane which preserves the angles between the curves passing through $z_0$ but cha
...[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) pp. Chapt. 2 (Translated from Russi
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$#C+1 = 12 : ~/encyclopedia/old_files/data/E036/E.0306950 Extended complex plane
The complex $ z $-
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...Cartesian coordinates $(a,b)$. The affix is sometimes identified with the complex number itself.
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$#C+1 = 29 : ~/encyclopedia/old_files/data/C024/C.0204150 Complex of lines
is called a ray of the complex. Through each point $ M $
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''semi-circular domain, with symmetry plane $\{z_n=a_n\}$''
A domain in the space of $n$ complex variables which, for each point $z=(z_1,\dots,z_{n-1},z_n)\equiv('z,z_n)$,
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The set of points in a plane between two parallel straight lines in this plane. The coordinates $ x , y $
in the complex plane $ ( z = x + iy ) $
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...mplex numbers too, the said functions can be extended to the whole complex plane and be studied for their own sake without any geometric application.
...igonometry]] the main problem is to compute three of the six elements of a plane triangle (3 sides and 3 angles) if three of them are known. The object of [
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...ce of the [[complete analytic function]]. For instance, in the cut complex plane $D = \mathbf{C} \setminus \{ z = x : -\infty < x \le 0 \}$ the multi-valued
...[2]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''2''' , Chelsea (1977) (Translated from Russian)</TD></TR>
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...[[complex number]]s, with $ab\neq0$. Two-term equations have $n$ distinct complex roots
The roots of a two-term equation in the complex plane are located on the circle with radius $|b/a|^{1/n}$ and centre at the coord
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variables $x_1,\dots,x_n$, real or complex. If $a=0$, the linear function is called
...ll variables $x_1,\dots,x_n$ and coefficients $a_1,\dots,a_n, a$ are real (complex) numbers,
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...ctive plane]] is an asymmetric variety, since the self-intersection of the complex straight line is $+1$ or $-1$, depending on the orientation. Certain knots
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onto which the extended complex plane $ \overline{\mathbf C}\; $
can be taken as the Riemann sphere and the plane $ \overline{\mathbf C}\; $
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''domain without large complex discs''
...complex disc. Extreme examples are the complex disc and the whole complex plane. The former is an example of a Kobayashi-hyperbolic manifold while the latt
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A transformation taking each point $A$ of the plane to the point $A'$ on the [[ray]] $OA$ for which $OA'.OA = k$, where $k$ is
and in the complex plane by the formula $z' = k / \bar z$. An inversion is an anti-conformal mapping
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''for a meromorphic function $f(z)$ in a domain $G$ of the complex $z$-plane''
...n that the image of $g$ under the mapping $w=f(z)$ is not dense in the $w$-plane. The strengthened version of Fatou's theorem in the theory of boundary prop
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$#C+1 = 32 : ~/encyclopedia/old_files/data/K055/K.0505240 Kernel of a complex sequence
The set of points in the extended complex plane that for a sequence $ \{ z _ {n} \} $
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...are topologically equivalent to a circle in the Euclidean plane, while the complex projective straight line is topologically equivalent to a two-dimensional s
In the plane over an arbitrary algebraic field, a straight line is an algebraic curve of
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A bounded simply-connected domain $G$ in the complex plane such that its boundary is the same as the boundary of the domain $G_\infty$
Let $G_n$ be a sequence of simply-connected domains in the complex plane. Suppose that each contains a fixed disc $D$ with centre $z_0$. Let
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