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Complex Differentiation Complex Analysis a.cyclohexane.molecule Complex Differentiation Two-dimensional complex functions have a two-dimensional complex derivative; the four-dimensional combination is difficult to visualize. The two-dimensional real approach of interpreting the derivative as the slope of a function is no longer possible; instead, we must look separately at the two complex planes on which lie the domain and codomain of a given function. Consider a mapping (function) f(z) = f(x + iy) = u(x; y) + iv(x; y), and an infinitesimal vector dz = hdx; dyi on the xy-plane. The mapping sends the vector to the corresponding vector d f(z) = f 0(z) dz = f 0(z) hdx; dyi = hdu; dvi in the uv-plane. Hence the action of the complex derivative f 0(z) is an expansion and rotation of the vector dz into the vector d f(z). Because we are working in the two-dimensional complex plane, infinitely many vectors of a given magnitude can emanate from the single point (x; y), differing only in their direction. Generally speaking, the effect of the complex derivative f 0(z) may differ for each of these vectors, expanding and rotating each by a different amount. We call analytic the rare function for which the complex derivative expands and rotates each of these infinitesimal vectors by the same amount, that is, for which the complex derivative is isotropic and path-independent. [We call entire those functions analytic on the entire complex plane; holomorphic those functions analytic on their domain of definition; and meromorphic those functions analytic, save for a set of isolated points, on their domain of definition.] The study of complex analysis deals primarily with analytic functions, for the requirement that the complex derivative f 0(z) expand and rotate each of these infinitesimal vectors by the same amount is a strong one, with far-reaching consequences. Furthermore, most of the functions with which we are familiar are analytic, and their derivatives the familiar ones from the calculus of real variables! Conformal Mapping Conformal mappings are those that preserve the magnitude and sense of angles; that is, two vectors a and b forming an angle θ going from a to b are mapped to two vectors f(a) and f(b) forming an angle θ going from f(a) and f(b). [As example of a non-conformal mapping, consider f(z) = z, which preserves the magnitude but not the sense of angles.] Based on the definition of analyticity, we readily claim that all analytic functions are conformal except at points z where f 0(z) = 0, conformality not being well-defined for the zero vector. The converse is trickier: is a conformal function analytic? Analyticity requires that the action of the derivative be isotropic. We know that conformal mappings are isotropic with regard to angles, but are they also isotropic with regard to magnitudes? Consider an infinitesimal triangle centered around z, mapped conformally to another infinitesimal triangle centered around f(z). Because a conformal mapping preserves angles we know that the two triangles are similar and hence that their lengths are proportional, suggesting isotropy with regard to magnitudes. For isolated points we can imagine that magnitudes may not be preserved (draw it!), but for planar regions we see the need for such isotropy. The preservation of angles over a region must imply a preservation of magnitudes: if a conformal mapping were anisotropic over a region, then its image is deformed and angles are no longer preserved, leading to a contradiction. Hence we claim that a conformal function having continuous partial derivatives (the requirement for conformality over a region instead of over isolated points) is analytic. Cauchy-Riemann Equations The Cauchy-Riemann equations are an algebraic representation of the path-independence of the derivative of analytic functions. In particular, we consider horizontal and vertical paths toward a point (x; y) for the function f(z) = f(x; y) = u(x; y) + iv(x; y): 0 0 f(x + ∆x; y) − f(x; y) f (z) = f (x; y) = lim = ux + ivx ∆x!0 ∆x 0 0 f(x; y + ∆y) − f(x; y) f (z) = f (x; y) = lim = vy − iuy ∆y!0 i∆y Hence equating real and imaginary parts gives us the Cauchy-Riemann equations, ux = vy uy = −vx Complex Analysis a.cyclohexane.molecule The Cauchy-Riemann equations provide another means of identifying an analytic function given its component functions u and v; alternatively, given one component u or v, the other can be determined up to an additive constant by finding its partial derivatives. Just as we claimed that conformal functions were analytic, so too must functions satisfying the Cauchy-Riemann equations be analytic (as long as their partial derivatives are continuous), conformality and the Cauchy-Riemann equations being two different representations of the same notion of isotropy. Further, note that the equivalence of partial derivatives on two paths is enough to guarantee analyticity and hence the equivalence of partial derivatives on all paths. We further make the observation that the component functions u and v of an analytic function are harmonic; that is, they satisfy the Laplace equation fxx + fyy = 0, as can be confirmed by suitable application of the Cauchy- Riemann equations. For a function f(z) = u(x; y) + iv(x; y), we call v the harmonic conjugate of u (though not the other way around) if u and v satisfy the Cauchy-Riemann equations. Harmonic functions, making use of conformal mapping, find applications in simplifying and solving boundary-value problems. Complex Integration Whereas on the real axis there is only one option for the path of integration between two points, on the complex plane we have infinitely many paths of integration, and we must work not with regular integrals but with contour integrals, path integrals in the complex plane over piecewise-smooth curves. Just as with multivariable functions, we may evaluate these path integrals by parametrization or by the fundamental theorem of line integrals, if the integrands are analytic (and thus have antiderivatives) over the path. [While non-analytic functions can still be path-integrated provided their partial derivatives are piecewise continuous, and while some such integrals are even significant, they are ultimately of less concern. An interesting example is the integral I I Z 2π Z 2π 2 Z 2π Z r −iθ iθ −iθ iθ r z¯dz = re d(re ) = re (ire ) dθ = 2i dθ = 2i r dr dθ = 2iAenc C C 0 0 2 0 0 which allows one to find Aenc, the area enclosed by the closed contour C.] Most complex functions share the same derivatives and antiderivatives as their real counterparts, but subtle issues arise when dealing with fractional exponents and logarithms, such functions being multi-valued and hence requir- ing the imposition of further restrictions before their antiderivatives can be well defined. Consider the complex logarithm log z = log[r exp(iθ)] = ln r + iθ, which has infinitely many possible values for θ. [If an angle θ satisfies the equation, so too does any integral multiple of θ + 2nπ.] In order for the necessarily single-valued path integral to make sense, we must make a branch cut, specifying an angular domain that bounds θ to an interval spanning at most 2π radians. Consider the closed contour C, a circle of radius r centered at the origin and once traversed in a positive (counterclockwise) sense, and a corresponding integral I dz r exp i(φ+2π) = [log(z)]r exp iφ = ln r + i(φ + 2π) − ln r − iφ = 2πi C z where we have implicitly defined a branch cut for which θ 2 (φ, φ + 2π), for otherwise log z would not have been analytic over the path and we would not have been able to use the fundamental theorem for path integrals. Note that the value of the integral is independent of r, and also that because neither of the angular bounds of the path integral are within the branch cut, we are actually dealing with a doubly improper integral. Nevertheless, because φ and φ + 2π are respectively the right-handed and left-handed limits of the domain defined by the branch cut, the integral may still be evaluated|and, perhaps surprisingly, the resulting integral is non-zero. [We may evaluate a similar integral in the same manner, where C0 now represents a circle of radius r centered around ρ: I dz ρ+r exp i(φ+2π) = [log(z − ρ)]ρ+r exp iφ = ln r + i(φ + 2π) − ln r − iφ = 2πi C0 z − ρ obtaining the same result as before. Both these integrals will be of great importance in upcoming sections.] Complex Analysis a.cyclohexane.molecule Cauchy-Goursat Theorem Green's theorem for line integrals applies equally well to contour integrals. In particular, I I I I f(z) dz = u(x; y) + iv(x; y) (dx + idy) = u(x; y) dx − v(x; y) dy + i v(x; y) dx + u(x; y) dy C C C C ZZ ZZ = ux − vy dx dy + i vx + uy dx dy S S = 0 where the second equality follows from application of the Cauchy-Riemann equations. In applying the Cauchy- Riemann equations we have made the implicit assumption that the function is analytic within and on the simply connected region|one in which every simple closed contour within it encloses only points within the region| bounded by C and also that the derivative of f is continuous; the latter assumption is actually unnecessary. In extending the theorem to multiply connected regions, we need only split up each multiply connected region with line segments to form multiple simply connected regions; the theorem then applies to each of these regions.
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