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Why Wirtinger Derivatives Behave So Well

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Why Wirtinger Derivatives Behave So Well Why Wirtinger derivatives behave so well Bart Michels∗ April 14, 2019 Contents 1 Introduction 2 2 Partial derivatives revisited 2 3 Complexification 3 3.1 Complexified differentials: two variables . .4 3.2 Complexified differentials: multiple variables . .4 3.3 Conjugation . .5 4 Chain rule 7 5 Holomorphic functions 10 6 Power series 11 7 Concluding remarks 12 ∗Universit´eParis 13, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse, France. Email: [email protected] 1 1 Introduction Given a smooth function f : C ! C, it makes a priori no sense to talk about @f @f the Wirtinger derivatives and . We could define them by @z @z¯ @f 1 @f @f = − i @z 2 @x @y (1) @f 1 @f @f = + i @z¯ 2 @x @y but that does not explain why we denote them like partial derivatives, where the confusing sign convention comes from, why they behave like partial derivatives, and why they act on power series in z and z in the way they should. Here follows a construction of Wirtinger derivatives that avoids the explicit formula (1). The reader may be interested in the definitions in section 3, in the sanity check in Proposition 3.2, the chain rule from Proposition 4.3 and the results from section 6. 2 Partial derivatives revisited It is worthwhile to take a second look at what a partial derivative is. Con- sider finite-dimensional real vector spaces V and W .1 They carry a canonical smooth structure. Consider a differentiable function f : V ! W . At every point p 2 V , f has a differential Dpf : V ! W , an R-linear map. In order to talk about the partial derivatives of f, we need to fix a basis of V , say ∗ ∗ ∗ (e1; : : : ; en), which has a dual basis, (e1; : : : ; en). Here ei is characterized by ∗ the property that ei (ej) = δi;j. We can then define @f ∗ (p) := (Dpf)(ei)(2) @ei n Usually, we take V = R and let (ei) be the standard basis, whose dual we are used to denote by (xi). The same holds for a holomorphic map between complex vector spaces, where the differential Dpf is then C-linear. We see that a partial derivative really is a directional derivative, and we need not fix an entire basis to define the derivative in the direction of ei. However, in some situations, we are given a linear form e∗ 2 V ∗. In order 1We may and will also take W to be a complex vector space and regard it as a real one, by restriction of scalars. 2 to define the derivative w.r.t. e∗, we have to extend it to a basis of V ∗, in order to make sense of the antidual vector e 2 V . We recall: Proposition 2.1. Let k be a field and V a finite dimensional k-vector space. ∗ Then for every basis (xi) of V there is a unique basis (ei) of V such that ∗ (xi) = (ei ). ◦ We denote the unique antidual basis of (xi) by (xi ). When k 2 fR; Cg and ∗ (xi) is a basis of V , we thus have, for a linear map A : V ! W , @A ◦ (3) = A(xi ) @xi Note that the choice of a basis of V ∗ is implicit in the LHS. ∗ Proposition 2.2. Let (xi) be a basis of V and f : V ! W an isomorphism. Then ◦ −1 ◦ f(xi ) = (xi ◦ f ) Proof. It suffices to remark that −1 ◦ (xj ◦ f )(f(xi )) = δi;j 3 Complexification Now let W be a complex vector space and consider a differentiable function 2 2 2 f : R ! W . Take p 2 R . Then f has a differential Dpf : R ! C, which is R-linear. We would like to make sense of (@f=@z)(p), so we want to view 2 z andz ¯ as linear forms on R that form a basis of its dual. The solution 2 is to complexify R . Definition 3.1. Let V be a real vector space. Then there exists a complex vector space VC and an R-linear map ι : V ! VC such that for every R-linear map f : V ! W to a complex vector space, there exists a unique C-linear map Cf : VC ! W such that f = (Cf) ◦ ι: f V W ι Cf VC We call (VC; ι) a complexification of V . 3 This is a special case of the adjunction between the extension of scalars and restriction of scalars functors. We may indeed take VC = V ⊗R C. We call VC the complexification of V . This gives a functor: for real vector spaces V1;V2 and a linear map f : V1 ! V2, there is a unique C-linear map fC :(V1)C ! (V2)C such that the following diagram commutes: f V1 V2 ι1 ι2 fC (V1)C (V2)C In particular, a complexification is unique up to unique C-isomorphism that is compatible with the maps ι. n n If we denote ι1 : R ! C the inclusion, then we have (R )C = C where the n n canonical map ιn : R ! C is ι1 × · · · × ι1. 3.1 Complexified differentials: two variables 2 We have the R-linear map Dpf : R ! C. Passing to the complexification 2 2 of R , this gives us a canonical C-linear map CDpf : C ! C. We want to apply this map to an appropriate vector, and call the result (@f=@z)(p). To 2 do that, we introduce coordinates on C . 2 We have R-linear maps z; z¯ : R ! C. They give canonical C-linear maps 2 Cz; Cz¯ : C ! C, which are given in coordinates by (Cz)(z1; z2) = z1 + iz2 ; (Cz¯)(z1; z2) = z1 − iz2 2 ∗ They form a basis of the dual (C ) , and we define, in analogy to (3): @f ◦ (p) := (CDpf)((Cz) ) (4) @z @f (p) := ( D f)(( z¯)◦) @z¯ C p C It is a straightforward calculation that these definitions coincide with (1). ◦ 1 i t ◦ 1 i t Explicitly, (Cz) = ( 2 ; − 2 ) and (Cz¯) = ( 2 ; 2 ) . But we don't like to compute, so we will instead derive those formulas from the chain rule from Proposition 4.2. 3.2 Complexified differentials: multiple variables n Take n > 1 and let f : C ! C be differentiable. We denote the coordinates n on C by (zi). We want to understand what @f=@zi means. 4 n Take p 2 C . As in the two variable case, we have a C-linear map CDpf : 2n 2n C ! C, and we have coordinates Czi; Cz¯i on C . We define @f (p) := ( D f)(( z )◦) @z C p C i (5) i @f ◦ (p) := (CDpf)((Cz¯i) ) @z¯i Using the universal property of complexification, one verifies that these def- n initions do not depend on the choice of complexification of C . So far this was just language. What we really want is to answer the questions from the introduction. For now, we remark the following: n Proposition 3.2. Write the standard complex coordinates on C as (zi). Then @zi @z¯i = = δi;j @zj @z¯j @z @z¯ i = i = 0 @z¯j @zj n Proof. The map zi : C ! C is R-linear, so its differential equals itself: Dp(zi) = zi. Thus the extended differential CDp(zi) equals the coordinate 2n Czi of C . Similarly forz ¯i. The statement now follows from the definition of @=@zi and @=@z¯i, and the definition of the antidual basis. ◦ ◦ Defining the partial derivatives in terms of the antidual vectors (Czi) ; (Cz¯i) has a great advantage. We need not know exactly what those vectors are, yet 2n we know the coordinates of each w 2 C in this basis: they are zi(w); z¯i(w). This observation will serve us when we prove chain rules, in the next section. 3.3 Conjugation The end goal of this subsection is Proposition 3.5. The remainder of the text does not rely on it, so it may safely be skipped. For every R-automorphism σ of C, a complexification VC has a canonical R- automorphism given by 1 ⊗ σ : V ⊗R C ! V ⊗R C, conjugated by the unique ∼ C-isomorphism VC = V ⊗R C compatible with ι. This allows us to define conjugation on VC. The conjugation may also be constructed intrinsically as follows. Define another C-linear structure on VC by precomposing the ring representation C ! EndZ(VC) with complex conjugation. Call the resulting -vector space V 0 . We can define conjugation as the unique - C C C homomorphism V ! V 0 compatible with ι, given by the universal property C C 5 of VC. It is not hard to see that, because conjugation is an automorphism of , we have that (V 0 ; ι) is still a complexification of V , which implies that C C the conjugation on VC is an automorphism. More generally, given commutative rings A; B, a ring homomorphism g : A ! B, an A-endomorphism σ of B and an A-module M, the endomorphism σ induces a canonical A-linear M-endomorphism of the extension of scalars MB = g∗M. It can be constructed in two ways, as before. We denote it by M ⊗ σ, because we may also view it as the image of σ under the functor M ⊗ −. The map f = M ⊗ σ is σ-semilinear: it satisfies, by construction, f(lm) = σ(l)f(m) for m 2 MB, l 2 L. If M; N; P are B-modules, f : M ! N is a σ-semilinear and g : N ! P is τ-semilinear, then g ◦ f is (τ ◦ σ)-semilinear.
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