Complex variables analytic functions

Conformal Mapping Every function of a complex variable w = f z) = u x, y) + iv(x, y) transforms the x, y plane into the u, v plane in some manner. A conformal transformation is one in which angles between curves are preserved in magnitude xnd sense. Every analytic function, except at those points where/ ( ) = 0, is a conformal transformation. See Fig. 3-48. 

These three equations (11), (12), and (13) contain three unknown variables, ApJt kn and sr The rest are known quantities, provided the potential-dependent photocurrent (/ph) and the potential-dependent photoinduced microwave conductivity are measured simultaneously. The problem, which these equations describe, is therefore fully determined. This means that the interfacial rate constants kr and sr are accessible to combined photocurrent-photoinduced microwave conductivity measurements. The precondition, however is that an analytical function for the potential-dependent microwave conductivity (12) can be found. This is a challenge since the mathematical solution of the differential equations dominating charge carrier behavior in semiconductor interfaces is quite complex, but it could be obtained,9 17 as will be outlined below. In this way an important expectation with respect to microwave (photo)electro-chemistry, obtaining more insight into photoelectrochemical processes... 

APPLIED COMPLEX VARIABLES, John W. Dettman. Step-by-step coverage of fundamentals of analytic function theory—plus lucid exposition of 5 important applications Potential Theory Ordinary Differential Equations Fourier Transforms Laplace Transforms Asymptotic Expansions. 66 figures. Exercises at chapter ends. 512pp. 5)4 x 8)4. 64670-X Pa. 10.95... 

These zeros uk of QK(u) coincide with the eigenvalues of both the evolution matrix U and the corresponding Hessenberg matrix H from Eqs. (131) and (130), respectively. The zeros of Qk(u) are called eigenzeros. The structure of CM is determined by its scalar product for analytic functions of complex variable z or u. For any two regular functions/(m) and g(u) from CM, the scalar product in CM is defined by the generalized Stieltjes integral ... 

For a complex function/(z) —/i(z) + if2(z) of the complex variable z, which is analytic in the upper half-plane of z and decays faster than z 1, the two Kramers-Kronig 26,127 relations are... 

It should be observed that since the operators T to be discussed include differential operators, it may be practical from the very beginning to assume that the functions F(Z) and G(Z) are analytic functions of all the complex variables involved. 

At this point it should be observed that even if it is natural to assume that all the functions F = F(Z) in the definition space A = F are analytic functions of the composite complex variable Z = zt, z2,..., zN, one has still to be very careful in the definition of the complex transformation U. Some of the complications which may occur are well illustrated by the simple example of complex scaling as defined in Eq. (2.26). 

For the sake of simplicity, we will consider an analytic function / = f(z) of a single complex variable z. It is well known that except for the integer functions and the meromorphic functions defined in the entire complex plane, such an analytic function usually has a natural domain restricted by a boundary of singularities, over which it cannot be analytically continued. This means that even in the case of complex scaling v = v(rj)—for certain values of 11—the transformation may take the variable rjz outside the domain of analyticity, and the operation is then meaningless. As an example, we may consider the analytic functions / = f(z) which are defined only within the unit circle z < 1, with this circle as natural boundary, and it is then evident that the complex scaling is meaningful only for 7 < 1. 

The analytical calculations involve determining the auxiliary functions of complex variables. Except for the simple shapes of WP surfaces, this presents some difficulties. Therefore, along with the analytical methods of solving the inverse ECM problem, numerical methods have evolved. Normally, in the numerical solution, the variables are changed in order to reduce the initial problem to a problem in the region with the known boundary. 

Elanigan, E. J., Complex Variables, Harmonic and Analytic Functions, Mineola, NY Dover Pubhcations, 1983. Folland, G. B., Introduction to Partial Differential Equations, Princeton, NJ Princeton University Press, 1995. Fowler, A. C., Mathematical Models in the Applied Sciences, Cambridge, UK Cambridge University Press,... 

A function F(z) is said to be meromorphic as a function of the complex variable z in some domain if F(z) is analytic in that domain apart from isolated poles of finite order (see, e.g., [372]). 

Gunning, R.C., Rossi, H. Analytic Functions of Several Complex Variables. Prentice Hall 1965... 

The Cauchy-Reimann conditions express the relationship that exists between the real and imaginary parts of an analytic function in the complex plane in differential form. In our case the complex variable 2 is the frequency ... 

We will make use of Cauchy s theorem, according to which the integral value of an analytical function does not change under deformation of an integration contour if it does not intersect singularities on the complex plane of variable to. It is clear that deforming the contour of integration in the upper half-plane (Im m > 0) exponent e with an increase of Im TO tends to zero. 

The analytic structure of the particle-hole Green s function (w) as a function of the complex variable w is governed by single poles and branch cuts... 

These conditions on the real and imaginary parts of a function w(x, y) must be fulfilled in order for u to be a function of the complex variable z. If, in addition, u and v have continuous partial derivatives with respect to x and y in some region, then w(z) in that region is an analytic function of z. In complex analysis, the term holomorphic function is often used to distinguish it from a real analytic function. 

This is the central result in the theory of complex variables. It states that the line integral of an analytic function around an arbitrary closed path in a simple connected region vanishes ... 

As last illustrated, some peculiar behavior patterns arise with complex variables, so care must be taken to insure that functions are well behaved in some rational sense. This property is called analyticity, so any function w = f(s) is called analytic within some two-dimensional region R if at all arbitrary points, say Sq, in the region it satisfies the conditions ... 

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 ). 

---