1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions and the principal branch is defined by 1 i - z w =Arc cot z =^1 / 2 7r - ---: Log-. - 2z z +z 305 (z "/= ±i ...

306. (a) w = *Vz (b) w = * y'r-a(_,-z---z-1.,.--:)(,--z ---z....,2 )-. -. ·-..,.( z---zs-:-) ( z; f z j, i f j) (c) w =*Vaz - z1 ...

Elementary Functions 307 Bibliography L. V. Ahlfors, Complex Analysis, 3rd. ed., McGraw-Hill, New York, 1979. L. V. Ahlfors and ...

6 Diffe1rentiation 6.1 THE CONCEPT OF THE DERIVATIVE. MONOGENIC AND ANALYTIC FUNCTIONS Definition 6.1 Let w = f(z) be a single-v ...

Differentiation Other notations for the derivative are w', Df(z), and Examples Let f(z) = z^2 , G = <C. In this case we have ...

310 Chapter6 y (^0) x Fig. 6.1 The concept of the derivative at a point may be modified by considering the function f defined on ...

Differentiation 311 1. f is analytic on E if there is an open set A ::J E and a function f 1 analytic on A such that Ji IE = f. ...

312 Chapter 6 1. If f(z) = c (a constant) in A, f'(z) = 0 for all z EA. 2. If f'(zo) and g^1 (z 0 ) exist for z 0 E A, we have: ...

Differentiation 313 the point wo has only one inverse image). But D.w ~ 0 as D.z ~ 0, and conversely. Choosing z E N 0 1(zo) we ...

314 fact, we have so that lim R = 0 16.zl->O IAzl Chapter6 and lim ..!..__ = 0 6.z->0 Az With the "little-oh" notation the ...

Differentiation 315 but this is not an infinitesimal with respect to l~zl, since e ~x~y l~zl = ~x2 + ~y2 has no limit as ~x ---t ...

316 Chapter6 = [u(x + b.x, y + b.y) - u(x, y + b.y)] + [u(x, y + b.y) - u(x, y)] = ux(x + .Ab.x, y + b.y)b.x + [u(x, y + b.y) - ...

Differentiation 317 so that It:/ ~zl --t 0 as l~zl --t 0. 6.5 THE CAUCHY-RIEMANN EQUATIONS If we write w = f(z) = u(x, y) + iv(x ...

318 Chapter^6 Hence, the partial derivatives u.,, uy, v.,, Vy exist at (x,y), and they satisfy the equations Ux =Vy, Uy= -v.,. c ...

Differentiation 319 As ~z ~ 0 the last two terms in (6.5-7) tend to zero. Therefore, f'( z ) = tl.z-+0 Ii Ill ~ ~w uz = u., + zv ...

320 Chapter^6 y 0 x Fig. 6,3, function F(t) = f(z(t)), a::=; t ::=; (3, is differentiable on [a,(J], and F'(t) = f'(z)z'(t) = 0, ...

Differentiation 321 Hence, under the foregoing assumptions, the components u and v of an analytic function on an open set A sati ...

322 Chapter6 every harmonic complex function f is analytic, since not any two solutions of the Laplace equation can be taken as ...

Differentiation 323 so that Uxx + Uyy = O, and u is harmonic (in C). Since Vx = -Uy = 2y, we get v = 2xy + F(y) (6.6-6) where F( ...

324 Chapter6 so that v is multiple-valued. If we choose the principal value of the argument in the subregion Ri = {z: z + Jzl =f ...