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294:{\displaystyle \mathbb {C} ^{n}=\underbrace {\mathbb {C} \times \mathbb {C} \times \cdots \times \mathbb {C} } _{n}.}
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216:{\displaystyle \mathbb {C} ^{n}=\left\{(z_{1},\dots ,z_{n})\mid z_{i}\in \mathbb {C} \right\}}
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516:-space is the target space for holomorphic coordinate systems on
380:. The real and imaginary parts of the coordinates set up a
376:
over the complex numbers, with componentwise addition and
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369:, a graphical representation of the complex line.
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547:Analytic functions of several complex variables
508:is the study of such holomorphic functions in
328:are the (complex) coordinates on the complex
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512:variables. More generally, the complex
504:in each complex coordinate separately.
496:A function on an open subset of complex
7:
14:
449:{\displaystyle \mathbb {R} ^{2n}}
365:, is not to be confused with the
482:{\displaystyle \mathbb {C} ^{n}}
406:{\displaystyle \mathbb {C} ^{n}}
354:{\displaystyle \mathbb {C} ^{2}}
85:{\displaystyle \mathbb {C} ^{n}}
500:-space is holomorphic if it is
372:Complex coordinate space is a
184:
152:
1:
125:with itself. Symbolically,
118:{\displaystyle \mathbb {C} }
45:) is the set of all ordered
584:
493:over the complex numbers.
332:-space. The special case
20:-tuples of complex numbers
568:Topological vector spaces
563:Several complex variables
506:Several complex variables
491:topological vector space
363:complex coordinate plane
63:. The space is denoted
35:complex coordinate space
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456:. With the standard
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422:real coordinate space
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378:scalar multiplication
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321:{\displaystyle z_{i}}
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16:Space formed by the
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458:Euclidean topology
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115:
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518:complex manifolds
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98:Cartesian product
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530:Coordinate space
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57:, also known as
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543:Gunning, Robert
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60:complex vectors
55:complex numbers
21:
12:
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5:
581:
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571:
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545:; Hugo Rossi,
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301:The variables
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494:
492:
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423:
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368:
367:complex plane
364:
361:, called the
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331:
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92:, and is the
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56:
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44:
42:
36:
33:
31:
26:
19:
546:
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419:-dimensional
416:
374:vector space
371:
362:
329:
102:complex line
93:
58:
46:
40:
38:
34:
32:-dimensional
29:
22:
17:
502:holomorphic
25:mathematics
557:Categories
536:References
413:with the
382:bijection
278:⏟
269:×
266:⋯
263:×
255:×
201:∈
188:∣
169:…
524:See also
39:complex
100:of the
96:-fold
51:tuples
43:-space
27:, the
489:is a
37:(or
384:of
223:or
53:of
23:In
559::
520:.
460:,
424:,
514:n
510:n
498:n
475:n
470:C
442:n
439:2
434:R
417:n
415:2
399:n
394:C
347:2
342:C
330:n
314:i
310:z
289:.
284:n
273:C
259:C
251:C
243:=
238:n
233:C
210:}
205:C
196:i
192:z
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180:n
176:z
172:,
166:,
161:1
157:z
153:(
149:{
145:=
140:n
135:C
112:C
94:n
78:n
73:C
49:-
47:n
41:n
30:n
18:n
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