Cauchy-Riemann equations

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In complex analysis, the Cauchy-Riemann equations are two partial differential equations that a complex function must satisfy in some region of the complex plane for it to be analytic in that region.[1][2] This has important implications for the differentiability of complex functions. If we express a complex number z in terms of two real numbers, 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,

The Cauchy-Riemann equations can then be expressed in terms of u(x,y) and v(x,y) as,[2]

They are named after Augustin-Louis Cauchy and Bernhard Riemann.

Derivation

When defining the derivative of a real function, it is done through a limit. The same method can be used for complex differentiation, defining it as,[1]

However we require the derivative and so this limit to be unique. But in the complex plane we can approach the limit in many different ways (e.g. along the real axis or the imaginary axis among others). These must all give the same result. If Δzx+iΔy and expressing f in terms of u(x,y) and v(x,y) as before then the limit becomes,

Consider the cases of Δz being purely real or purely imaginary, corresponding to taking the limit along wither the real or imaginary axes. If Δz is real then Δy is zero meaning,

If instead the case of Δz being imaginary (Δx is zero) is considered then,

For f to be differentiable at a point z these two must be identical. Comparing real and imaginary parts gives the Cauchy-Riemann equations,

References

  1. 1.0 1.1 K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering, Cambridge University Press, 3rd ed., 2006
  2. 2.0 2.1 Cauchy-Riemann Equations from mathworld.wolfram.com

See also