3.1. COMPLEX DERIVATIVES 89

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1 3.1. COMPLEX DERIVATIVES 89 Teorem (Cain rule) Let f : D D C and g : D C be olomorpic functions on te domains D and D respectively. Suppose tat bot f and g are C 1 -smoot. 10 Ten (g f) (z) g (f(z)) f (z), z D. Proof: ave z 0 D, write w 0 f(z 0 ). By te C 1 -smoot condition and Taylor Teorem, we so tat Ten f(z 0 + ) f(z 0 ) + f (z 0 ) + o(), and g(w 0 + ) g(w 0 ) g (w 0 ) + o(), g(f(z 0 + )) g(f(z 0 )) g(w 0 + f (z 0 ) + o()) g(w 0 ) g (w 0 ) ( f (z 0 ) + o() ) + o ( f (z 0 ) + o() ) g (w 0 )f (z 0 ) + o(). g(f(z 0 + )) g(f(z 0 )) g (w 0 )f (z 0 ) + o() g (w 0 )f (z 0 ) g (f(z 0 ))f (z 0 ). [Example] cain rule. Let f(z) sin(z 2 3z + 1). Ten f (z) cos(z 2 + 3z + 1) (2z 3) by te Analytic Function Must be Holomorpic So far we ave defined olomorpic functions. Wat are examples of olomorpic functions? Te following teorem says tat any analytic function must be olomorpic. In fact, later we sall see te converse is also true, namely, any olomorpic function must be analytic. Teorem Let f(z) n0 a n(z a) n be convergenet, a C, wit te radius of convergence R > 0. Ten f is a olomorpic function on (a, R) and f (z) na n (z a) n 1, z (a, R). 10 Tis C 1 -smoot condition will be dropped later by Goursat teorem.

2 90 CHAPTER 3. HOLOMORPHIC FUNCTIONS Claim 1: g(z) : na n(z a) n 1 as te radius of conver- Proof:(Optional reading) gence R. To prove, we assume a 0. Ten by Teorem 2.4.2, 1 R lim n ( n a n e limn 1 n ln an 1 lim e n n 1 ln an )( ) n 1 lim n n e limn 1 n 1 ln an lim n n a n+1 lim n n (n + 1) a n+1 1 R Here in te last second equality, we used te fact tat n n as n. Claim 1 is proved. It remains to sow: In fact, f(z + ) f(z) g(z)?? 0, as 0 ( ) (z + ) n z n a n nz n 1 n ( ) n a n z n v 1 v v n2 v2 n ( n a n z v) n v 1 v n2 v2 ( ) ( z + ) n z n a n n z n 1 ( ) ( z + ) a n r n n z n n z n 1 r n r n [ {( ) n ( ) n } 1 z + z M n r r r { ( 1 z + M r ( z + ) z ) r z Mr (r z ) 2 (r ( z + )) 0, as 0, f (z) g(z), z (0, R). (3.3) (we coose r wit z + < r < R) ( ) n 1 ] z r } r (r z ) 2 (we use geometric series)

3 3.1. COMPLEX DERIVATIVES 91 were M > 0 is a constant suc tat a n r n M, n. (3.3) is proved. Corollary f is analytic f is olomorpic. Remark: 1. We ll prove later, by Caucy s Integral Formula, tat f is olomorpic f is analytic. (3.4) 2. By te teory of power series, an analytic function as derivatives of any order, wile by te definition of olomorpic function, a olomorpic function is only required to ave te differentiation of first order. Tese two definitions are so different, but tey are equivalent. 3. Wile Caucy studied complex-valued functions f wit f exists, Weierstrass publised an extensive assay in 1876 of is systematic foundation of analysis of analytic functions. Tis very influential paper dealt wit te problem of representation of single-valued complex functions. Jacobian of olomorpic functions Let f(z) u(x, y) + iy(x, y) be a complex-valued function. We can regard f as a map from an open subset of R 2 to R 2, (x, y) (u, v), so tat its Jacobian is defined by were z x + iy. [ u x Jf(z) : det Te following result sows tat te Jacobian of a olomorpic function is always non negative. v x u y v y Proposition Let f be a olomorpic function. Ten Jf(z) f (z) 2. ]

4 92 CHAPTER 3. HOLOMORPHIC FUNCTIONS Proof: Jf(z) u v x y u v y x ( u) 2 ( u) 2 + f (z) 2, x y were we used te Caucy-Riemann Equations and te formula f (z) u x + i v x. Preserving Angle Property Let C be a non-singular curve given in parametric form φ : (α, β) C, t x(t) + iy(t). Suc curve can be understood as a vector-valued function φ : (α, β) R 2, t (x(t), y(t)) if we identify R 2 wit C. Suppose tat C is non singular (i.e., if φ (z) x (t) + iy (t) exists and 0). For eac t, if we regard φ(t) x(t) + iy(t) as a vector (x(t), y(t)), φ (t) x (t) + iy (t) can be regarded as te tangent vector (x (t), y (t)) of te curve C at te point φ(t). Te curve C at te point φ(t) as a tangent vector wose direction is determined by arg(φ (t)). Now suppose tat C is in a domain D C. Let f : D C be a olomorpic function. Ten we obtain te image curve f(c) given by te parametric function ψ : f φ : (α, β) C, t f(φ(t)). Ten By te cain rule (Teorem 3.1.5), ψ (t) f (φ(t))φ (t). arg(ψ (t)) arg(f (z)) + arg(φ (t)), mod(2π). (3.5) Terefore, te tangent vector of te curve f(c) at f(φ(t)) is canged from te tangent vector of te curve C at φ(t), and suc cange is caused by arg(f (φ(t))), wic is called te angle of rotation. [Example] Let f(z) cos z. Determine te angle of rotation given by arg(f (z)) at te points z 1 i, z 2 1 and z 3 π + i. Proof: f (z) sinz so tat f (z 1 ) f (i) sin i e 1 e i sin 1,

5 3.1. COMPLEX DERIVATIVES 93 f (z 2 ) f (1) sin 1, f (z 3 ) f (π + i) sin(π + i) ei(π+i) e i(π+i) eiπ e 1 e iπ e ( 1)e 1 ( 1)e i e e 1 2 i sin 1. Here sin z eiz e iz and sin x ex e x as in calculus. 2 Terefore, te angle of rotation is given by α 1 arg( i sin 1) π 2, α 2 arg( sin 1) π, α 3 arg(i sin 1) π 2.