Advanced Calculus fi..

548 Advanced Calculus, Fifth Editioncontinue to hold for complex z. A general algebraic identity is formed by replacingthe variables wl ,..., w, in an algebraic equation by functions f ~(z), ..., f,(z).Thus in the two examples given, one hasW: + W; - 1 = 0 (w, = sinz, w2 = COSZ),To prove identities such asezl . ez' = Zl+Z?e , (8.33)it may be necessary to apply Theorem 11 several times. (See Problems 4 and 5below.)It should be remarked that although eZ is written as a power of e, it is best not tothink of it as such. Thus ell2 has only one value, not two, as would a usual complexroot. To avoid confusion with the general power function, to be defined later, weoften write e = exp z and refer to e: as the exponential3~nction of z.To obtain the real and imaginary parts of sin z, we use the identitysin (2 I + z2) = sin z cos 22 + COS 21 sin 22,which holds, by the reasoning described above, for all complex zl and z2. Hencesin (x + iy) = sin x cos iy + cos x sin iy . Now from the definitions (Section 8. I),sinhy = -i siniy,cosh y = cos iy.1 HenceIsinz = sinxcoshy +icosxsinhy. (8.35)ISimilarly, we prove, as in (8.13),cosz = cosxcoshy -isinxsinhy,sinhz = sinhxcosy +icoshxsiny, (8.36)Icoshz = coshx cos y + i sinhx sin y. , 'It .re:Conformal mapping. A complex function w = f (z) can be con&dered as a mappingfrom the xy-plane to the uv-plane as in Section 2.7. In the case of an analyticfunction f (z) this mapping has a special property: It is a conformal mapping. By thiswe mean that two curves in the xy-plane, meeting at (xo, yo) at angle cr, correspondto two curves meeting at the corresponding point (uo, vo) at the same angle a! (invalue and in sense-positive or negative). This means that a small triangle in thexy-plane corresponds to a similar small (curvilinear) triangle in the uv-plane. (Theproperties described fail at the exceptional points where f '(z) = 0.) Furthermore,every conformal mapping from the xy-plane to the uv-plane is given by an analyticfunction. For a discussion of conformal mapping and its applications, see Sections8.20 to 8.27.