Cauchy-Riemann conditions

Definition A.10 (Region) A domain, together with or vnthout its boundary points, is a region. If the region contains all the domain and its boundary points, it is a closed region. 

The set A, with or without its boundary points, is a region whereas B is not because set B is not a domain. 

Definition A.11 (Anal3rtic Function) If a function f z) is differentiable at every point within a neighborhood cfzo, f z) is analytic at zq. 

Analytic functions are also called regular and holotnorphic. If f z) is analytic for all finite values of z, it is an entire function. If /(z) is not analytic at a point zq, the point Zo is a singular point. 

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

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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... 

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 ... 

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

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 ... 

It has a unique, finite derivative at Sq, which satisfies the Cauchy-Riemann conditions. 

The Cauchy-Riemann conditions are the essential properties for continuity of derivatives, quite apart from those encountered in real variables. To see these, write the general complex function... 

These are the Cauchy-Riemann conditions, and when they are satisfied, the derivative dw/ds becomes a unique single-valued function, which can be used in the solution of applied mathematical problems. Thus, the continuity property of a complex variable derivative has two parts, rather than the one customary in real variables. Analytic behavior at a point is called regular, to distinguish from nonanalytic behavior, which is called singular . Thus, points wherein analyticity breaks down are referred to as singularities. Singularities are not necessarily bad, and in fact their occurrence will be exploited in order to effect a positive outcome (e.g., the inversion of the Laplace transform ). 

If wCs) = s, prove that the function satisfies the Cauchy-Riemann conditions and find the region where the function is always analytic (i.e., regular behavior). 

We have now seen that the requirement that the two integrals be exact differentials is exactly the requirement that the Cauchy-Riemann conditions be satisfied. This means of course that the line integral... 

To do this, we return to the Cauchy-Riemann conditions given by Equations 4-5 and 4-6. Simple integration shows that the corresponding streamfunction is given by the expression... 

The relevant boundary conditions on Esingie-vaiued likewise found by superposition. From the Cauchy-Riemann conditions in Equations 4-5 and 4-6, we rewrite... 

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... 

Comparison of Equations 4-66 and 4-69 shows that 9p(x,y)/9x = 9 F(x,y)/9y, while Equations 4-65 and 4-68 show that 9p(x,y)/9y = -9 F(x,y)/9x. Hence, the Cauchy-Riemann conditions are identically satisfied, so that the streamfunction in Equation 4-67 is complementary to the vortex pressure in Equation 4-64. These results provide expressions for streamline tracing in the presence of distributed singularities. In general, as we will show in Chapter 5, the flow past (or from) an arbitrary entity can be represented by superpositions of line sources and vortexes, respectively, having strengths f(x) and g(x). The net pressure field... 

There is no typographical error in Equation 4-78 the m-l-1 applies to pressure only. Using an approach similar to that of Equations 4-72 and 4-73, we obtain Cauchy-Riemann conditions consistent with Discussion 4-2, that is,... 

Cauchy-Riemann conditions consistent with Equation 4-82 are 9p/9x = dVIdy and d ldy = - ST/Sx, exactly the same as earlier ones given in Equations 4-74 and 4-75. By an argument similar to that used above, consider the function... 

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. 

Define the complex potential for general gas flows with arbitrary m in anisotropic media, and verify that your Cauchy-Riemann conditions satisfy Equations 4-17 and 4-18. 

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