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An Introduction to Complex Differentials and Complex Differentiability

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An Introduction to Complex Differentials and Complex Differentiability An Introduction to Complex Differentials and Complex Differentiability Raphael Hunger Technical Report TUM-LNS-TR-07-06 2007 Technische Universitat¨ Munchen¨ Associate Institute for Signal Processing Prof. Dr.-Ing. Wolfgang Utschick Contents 1. Introduction 3 2. Complex Differentiability and Holomorphic Functions 4 3. Differentials of Analytic and Non-Analytic Functions 8 4. Differentials of Real-Valued Functions 11 5. Derivatives of Functions of Several Complex Variables 14 6. Matrix-Valued Derivatives of Real-Valued Scalar-Fields 17 Bibliography 20 2 1. Introduction This technical report gives a brief introduction to some elements of complex function theory. First, general definitions for complex differentiability and holomorphic functions are presented. Since non-analytic functions are not complex differentiable, the concept of differentials is explained both for complex-valued and real-valued mappings. Finally, multivariate differentials and Wirtinger derivatives are investigated. 3 2. Complex Differentiability and Holomorphic Functions Complex differentiability is defined as follows, cf. [Schmieder, 1993, Palka, 1991]: Definition 2.0.1. Let A ⊂ C be an open set. The function f : A ! C is said to be (complex) differentiable at z0 2 A if the limit f(z) − f(z ) lim 0 (2.1) ! z z0 z − z0 exists independent of the manner in which z ! z . This limit is then denoted by f 0(z ) = df(z) 0 0 dz z=z0 and is called the derivative of f with respect to z at the point z . 0 A similar expression for (2.1) known from real analysis reads as df(z) f(z + ∆z) − f(z) = lim ; (2.2) dz ∆z!0 ∆z where ∆z 2 C now holds. Note that if f is differentiable at z0 then f is continuous at z0. An equivalent, but geometrically more illuminating way to define the derivative follows from the linear approximation of f in the local vicinity of z0 [Palka, 1991]. Definition 2.0.2. Let A be an open set. The function f : A ! C is said to be (complex) differen- tiable at z0 2 A if there exists a complex-valued scalar g such that f(z) = f(z0) + g · (z − z0) + e(z; z0); (2.3) holds for every z 2 A and the function e(·; ·) satisfies the condition e(z; z ) lim 0 = 0: (2.4) ! z z0 z − z0 4 2. Complex Differentiability and Holomorphic Functions 5 The remainder term e(z; z0) in (2.4) obviously is o(jz − z0j) for z ! z0 and therefore g · (z − z0) dominates e(z; z0) in the immediate vicinity of z0 if g =6 0. Close to z0, the differentiable function 0 f(z) can linearly be approximated by f(z0) + f (z0)(z − z0). The difference z − z0 is rotated by 0 0 \f (z0), scaled by jf (z0)j and afterwards shifted by f(z0). The concept of a differentiability in a single point readily extends to differentiability in open sets. Definition 2.0.3. Let U ⊆ A be a nonempty open set. The function f : A ! C is called holomor- phic (or analytic) in U, if f is differentiable in z0 for all z0 2 U. Moreover, if f is analytic in the complete open domain-set A, f is a holomorphic (analytic) function. An interesting characteristic of a function f analytic in U is the fact that its derivative f 0 is analytic in U itself [Spiegel, 1974]. By induction, it can be shown that derivatives of all orders exist and are analytic in U which is in contrast to real-valued functions, where continuous derivatives need not be differentiable in general. However, basic properties for the derivative of a sum, product, and composition of two functions known from real-valued analysis remain inherently valid in the com- plex domain. Assume that f(z) and g(z) are differentiable at z0. Then, the following propositions hold: Proposition 2.0.1. The sum f + g is differentiable at z0 and 0 0 0 (f + g) (z0) = f (z0) + g (z0): (2.5) Proposition 2.0.2. The product fg is differentiable at z0 and 0 0 0 (fg) (z0) = f (z0)g(z0) + f(z0)g (z0): (2.6) f Proposition 2.0.3. If g(z0) =6 0, the quotient g is differentiable at z0 and 0 f f 0(z )g(z ) − f(z )g0(z ) (z ) = 0 0 0 0 : (2.7) g 0 g2(z ) 0 Proposition 2.0.4. If f is differentiable at g(z0), the composition f ◦ g is differentiable at z0 and 0 0 0 (f ◦ g) (z0) = f (g(z0))g (z0) (chain rule): (2.8) Complex differentiability is closely related to the Cauchy-Riemann equations [Lang, 1993]. A nec- essary condition for f being holomorphic in U requires the Cauchy-Riemann equations to be sat- isfied. 6 2. Complex Differentiability and Holomorphic Functions Theorem 2.0.1: Let f(z) = u(z) + jv(z) with u(z); v(z) 2 R and z = x + jy with x; y 2 R. In terms of x and y, the function f(z) can be expressed as F (x; y) = U(x; y) + jV (x; y) with U(x; y); V (x; y) 2 R. A necessary condition for f(z) being holomorphic in U is that the following system of partial differential equations termed Cauchy-Riemann-equations holds for every z = x + jy 2 U: @U(x; y) @V (x; y) @U(x; y) @V (x; y) = and = − : (2.9) @x @y @y @x Proof : According to Definition 2.0.3, f(z) is holomorphic in U if f(z) is differentiable at every z 2 U. Differentiability at z implies that the limit f(z + ∆z) − f(z) F (x + ∆x, y + ∆y) − F (x; y) lim = lim ∆z!0 ∆z ∆x!0 ∆x + j∆y z=x+jy ∆y!0 exists no matter which curve ∆z mo ves along when approaching zero, see Definition 2.0.1 and (2.2). Setting ∆z = ∆x + j∆y, two possible curves for ∆z ! 0 are considered. The first curve goes in the horizontal direction with ∆y = 0 and ∆x ! 0 yielding F (x + ∆x, y) − F (x; y) f 0(z = x + jy) = lim ∆x!0 ∆x U(x + ∆x, y) − U(x; y) V (x + ∆x, y) − V (x; y) = lim + j ∆x!0 ∆x ∆x @U(x; y) @V (x; y) = + j : @x @x The second curve goes in the vertical direction with ∆x = 0 and ∆y ! 0 yielding F (x; y + ∆y) − F (x; y) f 0(z = x + jy) = lim ∆y!0 j∆y U(x; y + ∆y) − U(x; y) V (x; y + ∆y) − V (x; y) = lim + j ∆y!0 j∆y j∆y @U(x; y) @V (x; y) = + : j@y @y As both expressions have to be the same for f(z) being holomorphic, (2.9) immediately follows as a necessary condition. 2 The next theorem provides conditions under which the Cauchy-Riemann equations are sufficient for f(z) being holomorphic. Theorem 2.0.2: If the partial derivatives of U(x; y) and V (x; y) with respect to x and y are con- tinuous, the Cauchy-Riemann equations are sufficient for f(z) being holomorphic. Proof : See [Spiegel, 1974]. 2 2. Complex Differentiability and Holomorphic Functions 7 In the following, we give examples for analytic functions and functions which are not analytic. Examples for analytic functions: • f(z) = zn f 0(z) = nzn−1 az + b ad − bc • f(z) = f 0(z) = cz + d (cz + d)2 0 1 • f(z) = ln(z) f (z) = z • f(z) = exp(az) f 0(z) = a exp(az) Examples for non-analytic functions: • f(z) = jzj2 • f(z) = <fzg • f(z) = =fzg • f(z) = z∗ 3. Differentials of Analytic and Non-Analytic Functions The total differential of the bivariate function F (x; y) associated to the univariate function f(z) via F (x; y) = U(x; y) + jV (x; y) = f(z)jz=x+jy reads as [Henrici, 1974] @F (x; y) @F (x; y) dF = dx + dy: (3.1) @x @y Of course, differentiability of F (x; y) with respect to x and y in the real sense has to be imposed for the existence of the differential dF in (3.1). This implies the differentiability of the real-valued functions U(x; y) and V (x; y) with respect to x and y. Rewriting (3.1) by means of F (x; y) = U(x; y) + jV (x; y) yields @U(x; y) @V (x; y) @U(x; y) @V (x; y) dF = dx + j dx + dy + j dy: (3.2) @x @x @y @y Making use of dz = dx + jdy; (3.3) dz∗ = dx − jdy; the two differentials dx and dy can be expressed via 1 dx = (dz + dz∗) 2 1 (3.4) dy = (dz − dz∗) : 2 j Inserting (3.4) into the differential expression dF in (3.1) and reordering the result leads to 1 @U(x; y) @V (x; y) @V (x; y) @U(x; y) dF = + + j − dz 2 @x @y @x @y (3.5) 1 @U(x; y) @V (x; y) @V (x; y) @U(x; y) + − + j + dz∗: 2 @x @y @x @y A major result can already be anticipated here. Proposition 3.0.1. The differential of any analytical function f(z) does not depend on the differ- ential dz∗. 8 3. Differentials of Analytic and Non-Analytic Functions 9 Proof : Since any analytical function f(z) satisfies the Cauchy-Riemann equations in (2.9), the factor in front of dz∗ in the second line of (3.5) is zero.
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