1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 345 or d dt u[x(t), y(t)] = 0 and d dt v[x(t),y(t)] = 0 for a :::; t :::; {3. These equations imply that u[x(t), ...

346 Chapter6 Next suppose that Then we have or lfzl^2 + l!zl^2 + fzfze^2 i^8 + fzf-ze-^2 i^8 = C^2 Letting il = fzfz, B = lfzl^2 ...

Differentiation 347 since (Dou)(Dov)' - (Dov)(Dou)' = J [Exercises 6.2, problem S(c)] and (Dou)^2 + (Dov)^2 = lf8(z)l^2 = p^2 • ...

348 Chapter6 Proof If fz = 0, we have f 0 (z) = fze-^2 i^8 '!-0, so that lf 0 (z)I = lf.zl and Argf 0 (z) ='I/;= w - 28 + 2k7r ( ...

Differentiation From the formulas for f z and fz in polar form we get e i..PJ z + e -i'l/Jf z -- f r -- Ur + ZVr. ir(ei'l/J fz - ...

350 Chapter6 Fig. 6.7 By eliminating the parameter () we obtain the equation of the graph of the Kasner circle in the form 1(-fz ...

Differentiation 351 Find the directional derivative of f(z) = (z + i)^2 + 2z at z = 1 + i in the direction of the arc given by ...

352 (a) w = zz, z = 2 + i (b) w = (z - 1)^2 .Z, z = i (c) w = zRez, z = 1 - i Chapter 6 *8. Let w = f(z) = u(x,y) + iv(x,y) E 'D ...

Differentiation 353 Prove that the function defined by (1) maps the deleted complex plane C - {O} onto the circle l w _ ac - bd ...

354 Chapter6 and v2 x +v2 y -/3 7 ab 1 = -U-x Vx+....;_u_y_V_y = u~ + u~ = r 2 J 2 = iJT ( d) Also, prove that 1 u2 + u2 + v2 + ...

Differentiation 355 In particular, if f is monogenic at z, then ( dw d( ) 8 = f' ( z) ( dz d( ) 8 Show that fz = 2 ~ 1 2 .rfMz ...

356 Chapter 6 It should be noted that a·t each point z the linear function T: C -t C defined by T(~z) = fz~z + fz~z is precisely ...

Differentiation 357 provided that l~zl is taken small enough, say l~zl < 6, in which case llfzl - lfzll - (11111+11121) > ...

358 in the open neighborhood A since at its center we have h(zo) = lf(zo) - w'I < e =^1 / 2 m so thatμ < %m also. However, ...

Differentiation^359 Examples 1. For f(z) = z^2 we have fz = 2z, fz = 0. Thus lfzl = lf.:I = 0 holds at z = 0 only. Clearly, the ...

360 Hence we have If z I = If z I -# 0 for every z such that lz+21 = lz+ll which gives z = -3/ 2 +iy,. -oo < y < +oo Also, ...

Differentiation 361 one-to-one in some neighborhood of every point of A'. Hence f: A -t f(A) is not one-to-one either. The condi ...

362 Chapter6 so that lfzl-:/= lfzl for all z. Hence the mapping is one-to-one at least locally. In fact, the function defines a ...

Differentiation 363 6.15 CONFORMAL MAPPINGS In this section we discuss some elementary properties of the mappings de- fined at a ...

Chapter6 are constants at a fixed point whenever f is monogenic at that point [and J'(z) f= 0 in referring to 1/J]. Let dz 1 and ...