Changes

In [[complex analysis]], the '''Cauchy-Riemann equations''' are two [[partial derivative|partial]] [[differential equation]]s that a [[complex number|complex]] [[function]] must satisfy in some region of the [[complex plane]] for it to be [[analytic]] in that region.<ref name=MathematicalMethods>K.F. Riley, M.P. Hobson, S.J. Bence, ''Mathematical Methods for Physics and Engineering'', Cambridge University Press, 3^rd ed., 2006</ref><ref name=wolfram>[http://mathworld.wolfram.com/Cauchy-RiemannEquations.html Cauchy-Riemann Equations] from mathworld.wolfram.com</ref> This has important implications for the [[differentiation|differentiability]] of complex functions. If we express a complex number ''z'' in terms of two [[real number]]s, ''x'' and ''y'', through ''z''=''x''+''iy'', then a complex function, ''f(z)'', can be expressed in terms of two functions of ''x'' and ''y'', ''u(x,y)'' and ''v(x,y)'', as,

In [[complex analysis]], the '''Cauchy-Riemann equations''' are two [[partial derivative|partial]] [[differential equation]]s that a [[complex number|complex]] [[function]] must satisfy in some region of the [[complex plane]] for it to be [[analytic]] in that region.<ref name=MathematicalMethods>K.F. Riley, M.P. Hobson, S.J. Bence, ''Mathematical Methods for Physics and Engineering'', Cambridge University Press, 3rd ed., 2006</ref><ref name=wolfram>[http://mathworld.wolfram.com/Cauchy-RiemannEquations.html Cauchy-Riemann Equations] from mathworld.wolfram.com</ref> This has important implications for the [[differentiation|differentiability]] of complex functions. If we express a complex number ''z'' in terms of two [[real number]]s, ''x'' and ''y'', through ''z''=''x''+''iy'', then a complex function, ''f(z)'', can be expressed in terms of two functions of ''x'' and ''y'', ''u(x,y)'' and ''v(x,y)'', as,

:<math>

:<math>

f(z) = u(x,y) + iv(x,y)

f(z) = u(x,y) + iv(x,y)

[[Category:Mathematics]]

[[Category:Mathematics]]

[[Category:Complex analysis]]

[[Category:Complex Analysis]]