Cauchy-Riemann equations

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... 

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 ... 

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. 

Taking the x derivative of the first Cauchy-Riemann equation and the y... 

Equating the real and imaginary parts of Eqs. (13.21) and (13.22), we again arrive at the Cauchy-Riemann equations (13.10). 

Using this severe requirement and choosing z — zq first purely real i.e., zq = xo + iyo, and z — x + iyo, and then purely imaginary, i.e., z = xo + iy, one obtains the Cauchy-Riemann equations... 

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. 

Note that the Cauchy-Riemann equations imply that, if the real part of a complex function is known, its imaginary part is determined to within a constant. This point is discussed later, from another viewpoint. 

---