1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 325 valid at least in Nr(zo). Taking derivatives on both sides of (6.6-8) with respect to z, we have f'(z) =u1 [ ...

326. Chapter^6 Note If in (6.6-10) we let z = zo we get f(zo) = u(xo,Yo) + iv(xo,yo) = 2u(xo,Yo) + C so that C = -u(x 0 , Yo)+iv ...

Differentiation Then we have and 8zf = fz =^1 Mfx - ify) az1=fz=^1 /2Ux + ify) fx = fz + Jz, fy = i(fz - fz) With this new notat ...

328 Chapter^6 and (6.7-7) may be written as df = f'(z)dz. This is the formula for the differential of a function at a point wher ...

Differentiation 329 Example If f(z) = lzl^4 = z^2 z^2 , we have 8f /8z = 2zz^2 and 8f /az = 2z^2 z. We now proceed to establish ...

330 Chapter^6 Theorem 6.9 If w = g(z) is of class 'D(A) and if ( = f(w) is defined and analytic in an open set B that contains t ...

Differentiation 331 (6.8-11) Similarly, af af aw1 af aw2. -=--+-- ay aw1 ay aw2 ay (6.8-12) Therefore, the usual rules of real a ...

332 Chapter6 (c) w = eaz (d) w = sinz (e) w = cosz (f) w = tanz (g) w = sinhz (h) w = coshz (i) w = tanhz (j) w =^1 / 5 ez(sin^2 ...

Differentiation 333 Dtanh-^1 z = 1 ~ z 2 (z-/= ±1) the value of the square root to be taken in the formulas of the first four ·r ...

334 Chapter^6 Show that each of the following functions is harmonic in some domain, then determine the corresponding harmonic c ...

Differentiation 335 and 14. Suppose that f = u +iv is analytic in a region G. Prove the following. (a) ( :x lf(z)I) 2 + ( :y lf( ...

336 Chapter^6 where 8u/8s stands for the directional derivative of u in the direction of the vector s, and similarly for the oth ...

Differentiation 337 u.,., - Uyy = 0. Show that W1 = Re u( z' z) and W2 = Im u( z, z) are solutions of Wxy = 0 in the same neighb ...

338 Chapter^6 is a harmonic conjugate function in R. ~et f(z) = u(r, B) + iv(r, B) E '.D(A), A open, z = rei^9 -/:-0. Show that ...

Differentiation 339 y Ci t t + Llt 13 0 x Fig. 6.4 Definition 6. 7 The directional derivative off at z in the direction of the a ...

340 Chapter^6 and the directional derivative of the given function at z 0 E 'Y* (in the direction of 'Y) is given by f' 'Y (z 0 ...

Differentiation 6.10 EXPRESSION OF THE DIRECTIONAL DERIVATIVE IN TERMS OF fz, f-z, AND 6 341 Suppose that w = f(z) = u +iv is of ...

342 Chapter 6 (96], equals fz except for the constant factor 2i, assuming f to be of class C(l)(A). For f continuous on A, C a s ...

Differentiation has argument () ± 71", so that f8±7r(z) = fz + fze-^2 i(fJ±7r) = f8(z) From formula (6.7-7), namely, we obtain d ...

344 Chapter^6 The angle 7/J = Arg f8( z) is called the distortion angle at z. Theorem 6.13 If a function has differentiable comp ...