Cauchy-Riemann

That Kn(z) is analytic in the lower half plane can be demonstrated by showing that Rn(z) obeys the Cauchy-Riemann conditions according to which if... 

For two-dimensional flows, such a relation was used by Schlichting (1939) without any proof and was later provided in Gaster (1962). But, this can be shown for general disturbance field by noting that lj is an analytic complex function of a and / . Therefore, one can use the Cauchy- Riemann equation valid for complex analytic functions and here, these are given by. 

If/(z) possesses a derivative at and at eveiy point in some neighborhood of Zo, enflz) is said to be analytic at Zo- If the Cauchy-Riemann equations are satisfied and... 

This function, F Q, is called a complex intensity of a plane field. It is an analytical function outside the sources, which vanishes at infinity because, according to equation (6.17), its real and imaginary parts, ReF = —Fx, ImF = F, satisfy the Cauchy-Riemann conditions ... 

The Cauchy-Riemann conditions describe the criteria under which a complex function is analytic. 

Theorem A.1 (Cauchy-Riemann Conditions) The complex function... 

The fact that electric potential tp x, y) is harmonic, which comes from (2) allows to of use theory of complex variables, for 2D potential problems. Introducing harmonic conjugate function y/ x, y), where both of them satisfy Cauchy-Riemann equations (CR), one has holomorphic function ... 

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. 

It also follows that if fi,z) is analytic, then the real and imaginary parts satisfy the Cauchy-Riemann equations, and it can be represented by a Taylor series in the neighborhood of Zq- A complex-valued function that is analytic in the whole complex plane is called an entire function. If a complex-valued function fails to be analytic at Zq but is analytic at every other point in the neighborhood of Zq. then Zq is said to be an isolated singular point off For example, 0 is an isolated singular point of/(z) = 1/z. 

These are the Cauchy-Riemann conditions. Functions that satisfy the Cauchy-Riemann conditions are called analytic functions. The Cauchy-Riemann conditions are satisfied by any analytic function and, hence, any of its successive derivatives. This property of analytic functions makes them useful in the solution of problems in two-dimensional elasticity. 

Considering Eqn. (3.14) and the operation on the real and imaginary parts of an analytic function /(z), one can see that by using the definition of derivatives and the Cauchy-Riemann conditions ... 

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