The Cauchy-Riemann differential equations

However, because of Eq. (B.8b) there is essentially an infinite number of ways to approach the point z by making Ar smaller. This nonuniqueness of the limit in Ekp (B.9) is a consequence of the fact that one is pursuing a path in the complex plane where an infinite degeneracy of such patlis exists. For the limit in Eq. (B.9) to exist it is necessary that the specific path Az — 0 in the complex plane be irrelevant. 

Let us consider two distinguished paths, one along the reed and the other one along the imaginary axis of the complex plane. The first of these is characterized by Az = Ax and Ay = 0 so that we have from Eqs. (B.8b) and (B.9) 

Along the second path in the complex plane we have Ax = 0 so that Az = iAy. By exactly the same reasoning one can then show that in this case 

According to our supposition the two tixpressions in Eqs. (B.IO) and (B.ll) must be equal. For this equality to be reached, the real and imaginary parts of both expressions to be equal w hich gives rise to the so-called Cauchy-Riemann differential equations 

Theorem B.l A function f z) defined over a domain B f) CC is analytic if its partial derivatives with respect to x andy exist and the Cauchy -Riemann differential equations are satisfied. 

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As f (2) IS analytic, the Cauchy-Riemann differential equations are satisfied so that each of the two integrals above vanishes identically regardless of the specific choice of the path C. q.e.d. 

A function /(z) differentiable in the above sense is said to be analytic at z = zq. The Cauchy-Riemann equations are necessary, but not sufficient for analyticity. 

Hint. The induced kinetic differential equation for x and 3 is a Cauchy-Riemann- (or Erugin-) system, therefore for z = x iy an easily solvable (separable) differential equation can be written down. 

To show that this is the correct solution, we need only differentiate it, and check that it satisfies the Cauchy-Riemann conditions in Equations 4-4 and 4-5. Straightforward differentiation shows that... 

Verify by direct differentiation that Equations 4-42 and 4-43, and 4-44 and 4-45, satisfy the Cauchy-Riemann conditions. Repeat this exercise for Equations 4-58 and 4-61, and 4-64 and 4-67. 

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