Question

Hirsch C. Numerical Computation of Internal and external Flows. Volume 1-Fundamentals of Comp... 125/ 696

P.3.1 Show that the system of Cauchy–Riemann equations ∂u ∂x + ∂v ∂y = 0 ∂v ∂x − ∂u ∂y = 0 is of elliptic nature

Answer 1

It is a bit intresting question

think that f(z) is holomorphic function

z = x + iy

f(x,y) = f(x+i y) = f(z)

u(x,y) = F (f(x,y))

v(x,y) = F' (f(x,y))

so f(x,y) = u(x,y) + i v(x,y)

we know by Cauchy–Riemann equations ∂u / ∂x + ∂v / ∂y = 0 and ∂v / ∂x − ∂u / ∂y = 0

ux = - vy

vx = uy

Actually this will lead to a system of decoupled elliptic equations.

uxx = -uyy

vxx = -vyy

So it is fair to say C-R equations are elliptical in nature

I know its bit difficult to understand. But its like this. If you need more support please do comment.

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