2188:
331:
2004:
38:
110:
1878:, conformal maps are the simplest and thus most common type of causal transformations. Physically, these describe different universes in which all the same events and interactions are still (causally) possible, but a new additional force is necessary to affect this (that is, replication of all the same trajectories would necessitate departures from
1663:, the original coordinate basis. Note that this application is not a contradiction to the fact that conformal mappings preserve angles, they do so only for points in the interior of their domain, and not at the boundary. Another example is the application of conformal mapping technique for solving the
1384:
Conformal mappings are invaluable for solving problems in engineering and physics that can be expressed in terms of functions of a complex variable yet exhibit inconvenient geometries. By choosing an appropriate mapping, the analyst can transform the inconvenient geometry into a much more convenient
1760:
Conformal maps are also valuable in solving nonlinear partial differential equations in some specific geometries. Such analytic solutions provide a useful check on the accuracy of numerical simulations of the governing equation. For example, in the case of very viscous free-surface flow around a
1345:
Applications of conformal mapping exist in aerospace engineering, in biomedical sciences (including brain mapping and genetic mapping), in applied math (for geodesics and in geometry), in earth sciences (including geophysics, geography, and cartography), in engineering, and in electronics.
1458:
radians, meaning that the corner of two planes is transformed to a straight line. In this new domain, the problem (that of calculating the electric field impressed by a point charge located near a conducting wall) is quite easy to solve. The solution is obtained in this domain,
909:
1717:) over a plane domain (which is two-dimensional), and is transformed via a conformal map to another plane domain, the transformation is also harmonic. For this reason, any function which is defined by a
2685:
Kolaei, Amir; Rakheja, Subhash; Richard, Marc J. (2014-01-06). "Range of applicability of the linear fluid slosh theory for predicting transient lateral slosh and roll stability of tank vehicles".
809:
it is biholomorphic. The two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative. In fact, we have the following relation, the
2514:
Calixto, Wesley
Pacheco; Alvarenga, Bernardo; da Mota, Jesus Carlos; Brito, Leonardo da Cunha; Wu, Marcel; Alves, Aylton José; Neto, Luciano Martins; Antunes, Carlos F. R. Lemos (2011-02-15).
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with determinant one). Some authors define conformality to include orientation-reversing mappings whose
Jacobians can be written as any scalar times any orthogonal matrix.
2149:
1795:. The latter are sometimes called Lorentz boosts to distinguish them from circular rotations. All these transformations are conformal since hyperbolic rotations preserve
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Warwick highlights this "new theorem of relativity" as a
Cambridge response to Einstein, and as founded on exercises using the method of inversion, such as found in
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566:, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or
1298:
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is quite clumsy to solve in closed form. However, by employing a very simple conformal mapping, the inconvenient angle is mapped to one of precisely
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which is one-to-one and holomorphic on an open set in the plane. The open mapping theorem forces the inverse function (defined on the image of
354:
2967:
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2903:
2798:
1963:
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In plane geometry there are three types of angles that may be preserved in a conformal map. Each is hosted by its own real algebra, ordinary
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semi-infinite wall, the domain can be mapped to a half-plane in which the solution is one-dimensional and straightforward to calculate.
2843:
1883:
2021:
Gemeinsame
Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometrie, 2
951:. However, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic.
1305:
shows that there are far fewer conformal maps in higher dimensions than in two dimensions. Any conformal map from an open subset of
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574:
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821:
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2632:"Estimation of intrinsic and extrinsic capacitances of graphene self-switching diode using conformal mapping technique"
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2943:
2831:
592:
For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally invertible
2878:
1414:, arising from a point charge located near the corner of two conducting planes separated by a certain angle (where
1309:
into the same
Euclidean space of dimension three or greater can be composed from three types of transformations: a
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2007:
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The complex conjugate of a Möbius transformation preserves angles, but reverses the orientation. For example,
1034:
1823:
955:
919:
1991:
1950:. The Student Mathematical Library. Vol. 9. Providence, Rhode Island: American Mathematical Society.
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383:
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1018:. Informally, this means that any blob can be transformed into a perfect circle by some conformal map.
330:
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Each four-dimensional solution could be inverted in a four-dimensional hyper-sphere of pseudo-radius
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581:. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a
184:
146:
1826:
and associated methods of images for spheres and inversion. As recounted by Andrew
Warwick (2003)
999:
969:
646:
2804:
2667:
2612:
2516:"Electromagnetic Problems Solving by Conformal Mapping: A Mathematical Operator for Optimization"
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2200:
1969:
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1814:
A larger group of conformal maps for relating solutions of
Maxwell's equations was identified by
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are conformal. The preservation of compass directions makes them useful in marine navigation.
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can be transformed by a conformal map and still remain governed by a potential. Examples in
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1922:– a conformal transformation of the upper half-plane onto the interior of a simple polygon
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17:
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2260:"Quasi-conformal mapping with genetic algorithms applied to coordinate transformations"
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if the pulled back metric is conformally equivalent to the original one. For example,
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1973:
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maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°.
2917:
1987:
3024:
3014:
2955:
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2238:
1886:
is different). It is often used to try to make models amenable to extension beyond
2094:
Cortijo, Vanessa; Alonso, Elena R.; Mata, Santiago; Alonso, José L. (2018-01-18).
3030:
2893:
2020:
1764:
For discrete systems, Noury and Yang presented a way to convert discrete systems
2283:
1822:(1910). Their training at Cambridge University had given them facility with the
1361:
1061:
762:
to a holomorphic function), it preserves angles but reverses their orientation.
375:
307:
126:
2655:
3027:
in different geometries without and with magnetic field by
Gerhard Brunthaler.
2706:
1769:
1765:
1322:
1030:
711:
2898:, CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages,
2663:
2600:
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2127:
2119:
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1928:– transformations that preserve volume (as opposed to angles) and orientation
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993:
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37:
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that can be used to examine the field of flow around a
Joukowsky airfoil.
109:
2468:
2382:
2187:
Alim, Karen; Armon, Shahaf; Shraiman, Boris I.; Boudaoud, Arezki (2016).
1891:
1879:
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640:
2542:
2333:
2323:
2306:
2305:
Berezovski, Volodymyr; Cherevko, Yevhen; Rýparová, Lenka (August 2019).
1753:. One example of a fluid dynamic application of a conformal map is the
603:
The notion of conformality generalizes in a natural way to maps between
2826:
Masters of theory : Cambridge and the rise of mathematical physics
2630:
Singh, Arun K.; Auton, Gregory; Hill, Ernie; Song, Aimin (2018-07-01).
1955:
1742:
1722:
765:
In the literature, there is another definition of conformal: a mapping
2783:
ASME 2020 International
Mechanical Engineering Congress and Exposition
2469:"Positional Accuracy Improvement Using Empirical Analytical Functions"
2448:"EAP - Proceedings of the Estonian Academy of Sciences – Publications"
2049:"Multipoint inverse airfoil design method based on conformal mapping"
1267:
805:) to be holomorphic. Thus, under this definition, a map is conformal
3036:
2895:
Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows
2072:
1890:, for example to permit description of the universe even before the
2424:"Mapping in a sentence (esp. good sentence like quote, proverb...)"
2205:
1804:
536:
387:
36:
1385:
one. For example, one may wish to calculate the electric field,
1434:
is the complex coordinate of a point in 2-space). This problem
3042:
1488:, and then mapped back to the original domain by noting that
41:
A rectangular grid (top) and its image under a conformal map
1741:, which is an approximation to fluid flow assuming constant
1285:
on a smooth manifold, as a class of conformally equivalent
2307:"Conformal and Geodesic Mappings onto Some Special Spaces"
2865:
Conformal invariants: topics in geometric function theory
2358:"Conformal Mapping of a Family of Real Conics on Another"
573:
The conformal property may be described in terms of the
2720:
Hinton, Edward; Hogg, Andrew; Huppert, Herbert (2020).
2258:
González-Matesanz, F. J.; Malpica, J. A. (2006-11-01).
1772:
through a well-know conformal mapping in geometry (aka
904:{\displaystyle (f^{-1}(z_{0}))'={\frac {1}{f'(z_{0})}}}
1911:– A conformal map extends continuously to the boundary
600:
sharply limits the conformal mappings to a few types.
1840:
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47:
2047:Selig, Michael S.; Maughmer, Mark D. (1992-05-01).
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2722:"Shallow free-surface Stokes flow around a corner"
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1048:Conformality with respect to three types of angles
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3010:Interactive visualizations of many conformal maps
2726:Philosophical Transactions of the Royal Society A
2962:(3rd ed.), New York: McGraw–Hill Book Co.,
2779:"A Pseudo S-plane Mapping of Z-plane Root Locus"
1864:Mathematical Theory of Electricity and Magnetism
1725:of equations defined by a potential include the
3031:Conformal Transformation: from Circle to Square
2362:Proceedings of the National Academy of Sciences
2473:Cartography and Geographic Information Science
2885:, Cambridge Tracts in Mathematics and Physics
1258:between two Riemannian manifolds is called a
355:
8:
2785:. American Society of Mechanical Engineers.
27:Mathematical function which preserves angles
1033:itself is conformal if and only if it is a
1022:Global conformal maps on the Riemann sphere
596:functions. In three and higher dimensions,
1807:. The introduction of translations in the
1791:which form a group including circular and
1325:, a conformal map may only be composed of
362:
348:
84:
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66:
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3025:Conformal Mapping images of current flow
2164:"7.1 GENETIC MAPS COME IN VARIOUS FORMS"
535:if it preserves angles between directed
1938:
277:
270:
192:
154:
116:
100:
3043:Joukowski Transform Interactive WebApp
2096:"Conformational Map of Phenolic Acids"
1948:Inversion Theory and Conformal Mapping
1064:. The conformal maps are described by
944:{\displaystyle z_{0}\in \mathbb {C} }
7:
2777:Noury, Keyvan; Yang, Bingen (2020).
2520:Mathematical Problems in Engineering
2467:López-Vázquez, Carlos (2012-01-01).
2100:The Journal of Physical Chemistry A
2025:Proceedings of the Imperial Academy
1854:in order to produce a new solution.
1803:) and the other rotations preserve
695:{\displaystyle f:U\to \mathbb {C} }
2923:, New York: McGraw–Hill Book Co.,
2919:Complex Variables and Applications
2867:, New York: McGraw–Hill Book Co.,
1689:
25:
2150:"Properties of Conformal Mapping"
1066:linear fractional transformations
962:, states that any non-empty open
958:, one of the profound results of
2002:
1319:special conformal transformation
455:{\displaystyle \mathbb {R} ^{n}}
329:
108:
1335:conformal linear transformation
390:, but not necessarily lengths.
3019:Wolfram Demonstrations Project
2687:Journal of Sound and Vibration
1710:{\displaystyle \nabla ^{2}f=0}
1630:
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1401:
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895:
882:
858:
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841:
825:
684:
478:
1:
2565:Leonhardt, Ulf (2006-06-23).
2356:Gronwall, T. H. (June 1920).
2223:10.1088/1478-3975/13/5/05lt01
3037:Online Conformal Map Grapher
1508:was obtained as a function (
1011:{\displaystyle \mathbb {C} }
981:{\displaystyle \mathbb {C} }
658:{\displaystyle \mathbb {C} }
2944:Encyclopedia of Mathematics
2832:University of Chicago Press
2567:"Optical Conformal Mapping"
2284:10.1016/j.cageo.2006.01.002
2264:Computers & Geosciences
1986:Richard M. Timoney (2004),
1946:Blair, David (2000-08-17).
1920:Schwarz–Christoffel mapping
1187:for some positive function
1072:In three or more dimensions
3089:
2189:"Leaf growth is conformal"
2019:Tsurusaburo Takasu (1941)
1353:
1080:
992:conformal map to the open
714:is everywhere non-zero on
528:{\displaystyle u_{0}\in U}
61:(bottom). It is seen that
32:Conformal (disambiguation)
29:
2960:Real and complex analysis
2863:Ahlfors, Lars V. (1973),
2707:10.1016/j.jsv.2013.09.002
1811:again preserves angles.
1779:
1643:, which is a function of
609:semi-Riemannian manifolds
579:coordinate transformation
272:Geometric function theory
218:Cauchy's integral formula
208:Cauchy's integral theorem
2937:E.P. Dolzhenko (2001) ,
2883:Conformal Representation
2822:Warwick, Andrew (2003).
2656:10.1088/2053-1583/aac133
2120:10.1021/acs.jpca.7b08882
2027:17(8): 330–8, link from
1374:stereographic projection
1356:Conformal map projection
1264:stereographic projection
811:inverse function theorem
487:{\displaystyle f:U\to V}
180:Cauchy–Riemann equations
18:Conformal transformation
2879:Constantin Carathéodory
2791:10.1115/IMECE2020-23096
2592:10.1126/science.1126493
2168:www.informatics.jax.org
2008:Geometry/Unified Angles
1988:Riemann mapping theorem
1888:curvature singularities
1824:method of image charges
1789:Lorentz transformations
1678:(that is, it satisfies
1636:{\displaystyle E(w(z))}
1380:Physics and engineering
956:Riemann mapping theorem
577:derivative matrix of a
386:that locally preserves
165:Complex-valued function
2746:10.1098/rsta.2019.0515
2485:10.1559/15230406393133
1992:Trinity College Dublin
1909:Carathéodory's theorem
1848:
1711:
1665:boundary value problem
1657:
1637:
1599:
1570:
1550:
1530:
1502:
1482:
1452:
1428:
1408:
1323:linear transformations
1281:One can also define a
1241:
1221:
1201:
1181:
1156:conformally equivalent
1148:
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1012:
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336:Mathematics portal
82:
75:
55:
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1727:electromagnetic field
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1202:
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1134:on a smooth manifold
1129:
1109:
1058:split-complex numbers
1035:Möbius transformation
1013:
983:
946:
906:
800:
780:
749:
729:
697:
660:
643:of the complex plane
634:
561:
559:{\displaystyle u_{0}}
530:
489:
457:
428:
408:
288:Augustin-Louis Cauchy
90:Mathematical analysis
76:
56:
40:
2383:10.1073/pnas.6.6.312
1926:Special linear group
1838:
1793:hyperbolic rotations
1685:
1647:
1609:
1598:{\displaystyle E(w)}
1580:
1560:
1540:
1520:
1492:
1481:{\displaystyle E(w)}
1463:
1451:{\displaystyle \pi }
1442:
1418:
1407:{\displaystyle E(z)}
1389:
1278:is a conformal map.
1231:
1211:
1191:
1180:{\displaystyle g=uh}
1162:
1138:
1118:
1098:
1000:
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920:
822:
789:
769:
738:
718:
672:
647:
623:
543:
506:
466:
437:
417:
397:
298:Carl Friedrich Gauss
233:Isolated singularity
175:Holomorphic function
65:
45:
30:For other uses, see
3063:Riemannian geometry
2989:"Conformal Mapping"
2939:"Conformal mapping"
2738:2020RSPTA.37890515H
2699:2014JSV...333..263K
2648:2018TDM.....5c5023S
2583:2006Sci...312.1777L
2577:(5781): 1777–1780.
2533:10.1155/2010/742039
2374:1920PNAS....6..312G
2324:10.3390/math7080664
2276:2006CG.....32.1432G
2215:2016PhBio..13eLT01A
2112:2018JPCA..122..646C
2065:1992AIAAJ..30.1162S
1882:motion because the
1860:James Hopwood Jeans
1816:Ebenezer Cunningham
1785:Maxwell's equations
1780:Maxwell's equations
1755:Joukowsky transform
1731:gravitational field
1370:Mercator projection
1283:conformal structure
1089:Riemannian geometry
1077:Riemannian geometry
598:Liouville's theorem
433:be open subsets of
393:More formally, let
185:Formal power series
147:Unit complex number
3058:Conformal mappings
3017:by Michael Trott,
2986:Weisstein, Eric W.
2913:Churchill, Ruel V.
1876:general relativity
1870:General relativity
1844:
1707:
1680:Laplace's equation
1653:
1633:
1595:
1566:
1546:
1526:
1498:
1478:
1448:
1424:
1404:
1333:, and is called a
1287:Riemannian metrics
1237:
1217:
1197:
1177:
1144:
1124:
1104:
1093:Riemannian metrics
1083:Conformal geometry
1008:
978:
941:
901:
795:
775:
744:
724:
692:
655:
629:
556:
525:
484:
452:
423:
403:
263:Laplace's equation
243:Argument principle
83:
71:
51:
2969:978-0-07-054234-1
2930:978-0-07-010855-4
2905:978-0-415-49271-3
2800:978-0-7918-8454-6
1965:978-0-8218-2636-2
1904:Biholomorphic map
1847:{\displaystyle K}
1828:Masters of Theory
1787:are preserved by
1774:inversion mapping
1751:irrotational flow
1674:If a function is
1656:{\displaystyle z}
1605:can be viewed as
1569:{\displaystyle z}
1549:{\displaystyle w}
1529:{\displaystyle E}
1501:{\displaystyle w}
1427:{\displaystyle z}
1299:classical theorem
1276:point at infinity
1274:augmented with a
1240:{\displaystyle u}
1220:{\displaystyle M}
1200:{\displaystyle u}
1147:{\displaystyle M}
1127:{\displaystyle h}
1107:{\displaystyle g}
1042:circle inversions
966:proper subset of
899:
798:{\displaystyle f}
778:{\displaystyle f}
747:{\displaystyle f}
727:{\displaystyle U}
632:{\displaystyle U}
615:In two dimensions
426:{\displaystyle V}
406:{\displaystyle U}
372:
371:
258:Harmonic function
170:Analytic function
156:Complex functions
142:Complex conjugate
74:{\displaystyle f}
54:{\displaystyle f}
16:(Redirected from
3080:
2999:
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2270:(9): 1432–1441.
2255:
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2246:
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2193:Physical Biology
2184:
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2091:
2085:
2084:
2059:(5): 1162–1170.
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1994:
1984:
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1977:
1956:10.1090/stml/009
1943:
1853:
1851:
1850:
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1797:hyperbolic angle
1768:into continuous
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1431:
1430:
1425:
1413:
1411:
1410:
1405:
1368:, including the
1364:, several named
1303:Joseph Liouville
1249:conformal factor
1246:
1244:
1243:
1238:
1226:
1224:
1223:
1218:
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1017:
1015:
1014:
1009:
1007:
987:
985:
984:
979:
977:
964:simply connected
960:complex analysis
950:
948:
947:
942:
940:
932:
931:
910:
908:
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664:
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654:
638:
636:
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594:complex analytic
565:
563:
562:
557:
555:
554:
534:
532:
531:
526:
518:
517:
500:angle-preserving
493:
491:
490:
485:
461:
459:
458:
453:
451:
450:
445:
432:
430:
429:
424:
412:
410:
409:
404:
364:
357:
350:
334:
333:
318:Karl Weierstrass
313:Bernhard Riemann
303:Jacques Hadamard
132:Imaginary number
112:
102:Complex analysis
96:
94:Complex analysis
85:
80:
78:
77:
72:
60:
58:
57:
52:
21:
3088:
3087:
3083:
3082:
3081:
3079:
3078:
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3068:Map projections
3048:
3047:
3006:
2984:
2983:
2970:
2954:
2936:
2931:
2911:
2906:
2888:
2862:
2859:
2857:Further reading
2854:
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2816:
2801:
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2714:
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2157:
2148:
2147:
2143:
2093:
2092:
2088:
2073:10.2514/3.11046
2046:
2045:
2041:
2018:
2014:
2001:
1997:
1985:
1981:
1966:
1945:
1944:
1940:
1935:
1915:Penrose diagram
1900:
1872:
1836:
1835:
1782:
1688:
1683:
1682:
1669:liquid sloshing
1645:
1644:
1607:
1606:
1578:
1577:
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1538:
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1518:
1517:
1490:
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1440:
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1416:
1415:
1387:
1386:
1382:
1366:map projections
1358:
1352:
1343:
1307:Euclidean space
1295:
1293:Euclidean space
1229:
1228:
1227:. The function
1209:
1208:
1189:
1188:
1160:
1159:
1136:
1135:
1116:
1115:
1096:
1095:
1085:
1079:
1074:
1054:complex numbers
1050:
1024:
998:
997:
968:
967:
923:
918:
917:
885:
874:
873:
857:
844:
828:
820:
819:
787:
786:
767:
766:
756:antiholomorphic
736:
735:
716:
715:
670:
669:
645:
644:
621:
620:
617:
583:rotation matrix
546:
541:
540:
509:
504:
503:
464:
463:
440:
435:
434:
415:
414:
395:
394:
368:
328:
238:Residue theorem
213:Local primitive
203:Zeros and poles
118:Complex numbers
88:
63:
62:
43:
42:
35:
28:
23:
22:
15:
12:
11:
5:
3086:
3084:
3076:
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3060:
3050:
3049:
3046:
3045:
3040:
3034:
3028:
3022:
3015:Conformal Maps
3012:
3005:
3004:External links
3002:
3001:
3000:
2981:
2968:
2952:
2934:
2929:
2909:
2904:
2886:
2876:
2858:
2855:
2852:
2851:
2845:978-0226873756
2844:
2814:
2799:
2769:
2712:
2693:(1): 263–282.
2677:
2622:
2557:
2506:
2479:(3): 133–139.
2459:
2439:
2415:
2368:(6): 312–315.
2348:
2297:
2250:
2179:
2155:
2141:
2106:(2): 646–651.
2086:
2039:
2029:Project Euclid
2012:
1995:
1979:
1964:
1937:
1936:
1934:
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1930:
1929:
1923:
1917:
1912:
1906:
1899:
1896:
1871:
1868:
1856:
1855:
1843:
1809:Poincaré group
1805:circular angle
1781:
1778:
1739:potential flow
1735:fluid dynamics
1706:
1703:
1700:
1695:
1691:
1652:
1632:
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1397:
1394:
1381:
1378:
1354:Main article:
1351:
1348:
1342:
1339:
1294:
1291:
1256:diffeomorphism
1247:is called the
1236:
1216:
1196:
1176:
1173:
1170:
1167:
1143:
1123:
1103:
1078:
1075:
1073:
1070:
1068:in each case.
1049:
1046:
1028:Riemann sphere
1023:
1020:
1006:
976:
939:
935:
930:
926:
914:
913:
912:
911:
897:
892:
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877:
872:
867:
863:
860:
856:
851:
847:
843:
838:
835:
831:
827:
807:if and only if
794:
774:
743:
723:
704:if and only if
690:
686:
683:
680:
677:
653:
628:
616:
613:
553:
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516:
512:
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344:
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340:
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338:
323:
322:
321:
320:
315:
310:
305:
300:
295:
293:Leonhard Euler
290:
282:
281:
275:
274:
268:
267:
266:
265:
260:
255:
250:
245:
240:
235:
230:
228:Laurent series
225:
223:Winding number
220:
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205:
197:
196:
190:
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177:
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2961:
2957:
2956:Rudin, Walter
2953:
2950:
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2649:
2645:
2642:(3): 035023.
2641:
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2199:(5): 05LT01.
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1961:
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1897:
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1893:
1889:
1885:
1884:metric tensor
1881:
1877:
1869:
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1831:
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1825:
1821:
1820:Harry Bateman
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1308:
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1300:
1292:
1290:
1288:
1284:
1279:
1277:
1273:
1269:
1265:
1261:
1260:conformal map
1257:
1252:
1250:
1234:
1214:
1194:
1174:
1171:
1168:
1165:
1157:
1141:
1121:
1101:
1094:
1090:
1084:
1076:
1071:
1069:
1067:
1063:
1059:
1055:
1047:
1045:
1043:
1038:
1036:
1032:
1029:
1026:A map of the
1021:
1019:
995:
991:
965:
961:
957:
952:
933:
928:
924:
890:
886:
878:
875:
870:
865:
861:
849:
845:
836:
833:
829:
818:
817:
816:
815:
814:
812:
808:
792:
772:
763:
761:
757:
741:
721:
713:
709:
705:
702:is conformal
681:
678:
675:
668:
642:
626:
614:
612:
610:
606:
601:
599:
595:
590:
588:
584:
580:
576:
571:
569:
551:
547:
538:
522:
519:
514:
510:
502:) at a point
501:
497:
481:
475:
472:
469:
462:. A function
447:
420:
400:
391:
389:
385:
381:
380:conformal map
377:
365:
360:
358:
353:
351:
346:
345:
343:
342:
337:
332:
327:
326:
325:
324:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
285:
284:
283:
280:
276:
273:
269:
264:
261:
259:
256:
254:
253:Schwarz lemma
251:
249:
248:Conformal map
246:
244:
241:
239:
236:
234:
231:
229:
226:
224:
221:
219:
216:
214:
211:
209:
206:
204:
201:
200:
199:
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191:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
162:
161:
160:
157:
153:
148:
145:
143:
140:
138:
137:Complex plane
135:
133:
130:
128:
125:
124:
123:
122:
119:
115:
111:
107:
106:
103:
99:
95:
91:
87:
86:
68:
48:
39:
33:
19:
2992:
2959:
2942:
2918:
2894:
2882:
2864:
2825:
2817:
2782:
2772:
2729:
2725:
2715:
2690:
2686:
2680:
2639:
2636:2D Materials
2635:
2625:
2574:
2570:
2560:
2543:10316/110197
2523:
2519:
2509:
2476:
2472:
2462:
2451:. Retrieved
2442:
2431:. Retrieved
2427:
2418:
2365:
2361:
2351:
2334:11012/188984
2314:
2310:
2300:
2267:
2263:
2253:
2242:. Retrieved
2196:
2192:
2182:
2171:. Retrieved
2167:
2158:
2144:
2103:
2099:
2089:
2056:
2053:AIAA Journal
2052:
2042:
2015:
2010:at Wikibooks
1998:
1982:
1947:
1941:
1873:
1863:
1857:
1827:
1813:
1783:
1763:
1759:
1673:
1509:
1435:
1383:
1359:
1344:
1341:Applications
1296:
1282:
1280:
1259:
1253:
1248:
1155:
1086:
1062:dual numbers
1051:
1039:
1025:
953:
915:
764:
618:
602:
591:
572:
499:
495:
392:
379:
373:
247:
194:Basic theory
93:
2890:Chanson, H.
2834:. pp.
2526:: e742039.
2311:Mathematics
1818:(1908) and
1514:composition
1362:cartography
1350:Cartography
1154:are called
708:holomorphic
641:open subset
376:mathematics
308:Kiyoshi Oka
127:Real number
3052:Categories
2453:2022-08-22
2433:2022-08-22
2317:(8): 664.
2244:2022-08-22
2206:1611.07032
2173:2022-08-22
1933:References
1799:, (called
1770:root locus
1766:root locus
1733:, and, in
1671:in tanks.
1081:See also:
712:derivative
605:Riemannian
587:orthogonal
494:is called
2994:MathWorld
2949:EMS Press
2809:234582511
2672:117531045
2664:2053-1583
2601:0036-8075
2552:1024-123X
2501:123894885
2493:1523-0406
2392:0027-8424
2343:2227-7390
2292:0098-3004
2128:1520-5215
2081:0001-1452
1974:118752074
1862:textbook
1747:viscosity
1719:potential
1690:∇
1576:, whence
1446:π
1327:homothety
1311:homothety
1270:onto the
994:unit disk
990:bijective
988:admits a
934:∈
834:−
760:conjugate
685:→
665:, then a
568:curvature
520:∈
496:conformal
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