8/19/2019 Cauchy Riemann

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DERIVING THE CAUCHY-RIEMANN EQUATIONS

IKJYOT SINGH KOHLI

In this brief document, we will derive the famous Cauchy-Riemann equations of complex analysis.

Consider a function of a complex variable,  z , where  z  =  x + iy, such that

(1)   f (z ) = u(z ) + iv(z ) = u(x + iy) + iv(x + iy),

where u  and  v  are real-valued functions. An  analytic  function is one that is express-ible as a power series in  z . That is,

(2)   f (z ) =

∞

n=0

anz n, an  ∈ C.

Then,

(3)   u(x + iy) + iv(x + iy) =∞

n=0

an(x + iy)n.

We formally differentiate this equation as follows. First, differentiating with respectto  x, we obtain

(4)   ux + ivx =∞

n=1

nan (x + iy)n−1

.

Differentiating with respect to  y, we obtain

(5)   uy + ivy  = i∞

n=1

nan (x + iy)n−1

.

Multiplying the latter equation by  −i  and equating to the first result, we obtain

(6)   −iuy +  vy  =∞

n=1

nan (x + iy)n−1 = ux + ivx.

Comparing imaginary and real parts of these equations, we obtain

(7)   ux =  vy, uy  =  −vx   ,

which are the famous  Cauchy-Riemann equations .

Date : October 14, 2015.

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