Cauchy-Riemann differential equation

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. 

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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