Editing Conformal map (section)

===Physics and engineering===
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 one. For example, one may wish to calculate the electric field, <math>E(z)</math>, arising from a point charge located near the corner of two conducting planes separated by a certain angle (where <math>z</math> is the complex coordinate of a point in 2-space). This problem ''per se'' 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 <math>\pi</math> 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, <math>E(w)</math>, and then mapped back to the original domain by noting that <math>w</math> was obtained as a function (''viz''., the [[function composition|composition]] of <math>E</math> and <math>w</math>) of <math>z</math>, whence <math>E(w)</math> can be viewed as <math>E(w(z))</math>, which is a function of <math>z</math>, 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 [[boundary value problem]] of [[Slosh dynamics|liquid sloshing]] in tanks.<ref>{{Cite journal|last1=Kolaei|first1=Amir|last2=Rakheja|first2=Subhash|last3=Richard|first3=Marc J.|date=2014-01-06|title=Range of applicability of the linear fluid slosh theory for predicting transient lateral slosh and roll stability of tank vehicles|journal=Journal of Sound and Vibration|volume=333|issue=1|pages=263–282|doi=10.1016/j.jsv.2013.09.002|bibcode=2014JSV...333..263K}}</ref>

If a function is [[harmonic function|harmonic]] (that is, it satisfies [[Laplace's equation]] <math>\nabla^2 f=0</math>) 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 [[potential]] can be transformed by a conformal map and still remain governed by a potential.  Examples in [[physics]] of equations defined by a potential include the [[electromagnetic field]], the [[gravitational field]], and, in [[fluid dynamics]], [[potential flow]], which is an approximation to fluid flow assuming constant [[density]], zero [[viscosity]], and [[irrotational vector field|irrotational flow]].  One example of a fluid dynamic application of a conformal map is the [[Joukowsky transform]] that can be used to examine the field of flow around a Joukowsky airfoil.

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 semi-infinite wall, the domain can be mapped to a half-plane in which the solution is one-dimensional and straightforward to calculate.<ref>{{cite journal |first1=Edward |last1=Hinton |first2=Andrew |last2=Hogg |first3=Herbert |last3=Huppert |year=2020 | title=Shallow free-surface Stokes flow around a corner | journal=Philosophical Transactions of the Royal Society A | volume=378 |issue=2174 |doi=10.1098/rsta.2019.0515|pmid=32507085 |pmc=7287310|bibcode=2020RSPTA.37890515H }}</ref>

For discrete systems, Noury and Yang presented a way to convert discrete systems [[root locus]] into continuous [[root locus]] through a well-know conformal mapping in geometry (aka [[Inversive geometry|inversion mapping]]).<ref>{{cite book |first1=Keyvan |last1=Noury |first2=Bingen |last2=Yang |year=2020 |chapter=A Pseudo S-plane Mapping of Z-plane Root Locus |chapter-url=https://www.researchgate.net/publication/343084262 |title=ASME 2020 International Mechanical Engineering Congress and Exposition |publisher=American Society of Mechanical Engineers |doi=10.1115/IMECE2020-23096|isbn=978-0-7918-8454-6 |s2cid=234582511 }}</ref>