1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

164 CHAPTER 5 • ELEMENTARY FUNCTIONS The domain for the function Log is the set of all nonzero complex numbers in the z plane, a ...

5.2 • THE COMPLEX LOGARITHM 165 When z = x + iO, where x is a positive rea.1 number, the principal value of the complex logarith ...

166 CHAPTER 5 • ELEMENTARY FUNCTIONS Our next result explains why Log (z 1 z2) = Log (z1) + Log (z2) didn't hold for the particu ...

5.2 • THE COMPLEX LOGARITHM 167 y v i(a+ 2ir) w= log,jz) z=ew ia Figure 5.4 The branch w = log 0 ( z) of tbe logru:ithm. of 8 E ...

168 CHAPTER 5 • ELEMENTARY FUNCTIONS J• J• Figure 5.5 T he Riemann surface for mapping w =log (z). The Riemann surface for the ...

(e) log (-3). (f) log8. (g) log ( 4i). (h) log (- ./3 -i). 5.2 • THE COMPLEX LOGARITHM 169 Use the properties of arg (z) in Sec ...

170 CHAPTER 5 • ELEMENTARY FUNCTIONS 10. For what values of z is it true t hat (a) Log ( *) =Log (z1) - Log (z2)? Why? (b} f;Log ...

5. 3 • COMPLEX EXPONENTS 171 Definition 5.4: Complex exponent Let c be a complex number. We define zc as zc =exp [clog (z)]. (5- ...

172 CHAPTER 5 • ELEMENTARY FUNCTIONS Identity (5-22) yields the principal values of JI+i and ii: J1+i=(l+i)~ =exp [~Log(l + i)] ...

5.3 • COMPLEX EXPONENTS 173 Hence Equation (5-21) becomes zf =exp [~log(z)] (5-23) [ 1 .8+2mr] =exp kln(r)+i k = rtexp(/+:mr) =r ...

1 74 CHAPTER 5 • ELEMENTARY FUNCTIONS )' 3 •^2 • I • • • • • -2 -I • • • • 2 3 • • -I • • -2 Figure 5. 6 Some of the values ...

5.3 • COMPLEX EXPONENTS 175 If we restrict zc to the principal branch, zC = exp [cLog (z)], then Equation (5-28) can be written ...

176 CHAPTER. 5 • ELEMENTARY FUNCTIONS (b) z"' (a a real number) is given by the equation z"' = r"' cos a9 + ir" sin a9, where r ...

5.4 • TR!GONOMETRJC AND HYPERBOLIC FUNCTIONS 177 Definition 5 .6: Trigonometric funct ions sin z cosz 1 1 tanz = --, cotz = -.-, ...

178 CHAPTER, 5 • ELEMENTARY FUNCTIONS For all complex numbers z for which the expressions are defined, d dz tan z = sec 2 z, d ...

5.4 • TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 179 where coshy = e• + 2 • - • and sinhy = ••7-•, respectively, are the hyperbolic ...

180 CHAPTER 5 • ELEMENTARY FUNCTIONS We demonstrate that oos (z 1 + z 2 } = oosz 1 cos z 2 - sin z 1 sin z 2 by making use of Id ...

5.4 • TRIGONOMETRIC AND HYPERBOLIC F UNCT IONS 181 y ) 2 I -K 2 1 -2 3 .1!. 2 ~·=sint x Figure 5.7 Vertical segments mappe ...

182 CHAPTER 5 • ELEMENTARY FUNCTIONS EXAMPLE 5.10 Find the values of z for which cos z = cosh 2. Solution Starting with Identi ...

5.4 • TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 183 y 3 2 I ' _,, 4 I -2 3 L 4 "' ..... -- x - I I Figure 6.8 Vertical segments ...