1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

184 CHAPTER 5 • ELEMENTARY FUNCTIONS The derivatives of the hyperbolic functions follow the same rules as in cal- culus: ! coshz ...

5.4 • TRlCONOMETRIC AND HYPERBOLIC FUNCTIONS 185 Ee ~ I '·{ c R L E(t) ~ O'\.... Figure 5 .9 An LRC circuit. The voltages EL, ER ...

186 CHAPTER 5 • ELEMENTARY FUNCTIONS The complex quantity Z defined by Z = R + i ( wL - w~) is called the complex impedance. Sub ...

5.4 • TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 187 (h) cosh if. (i) cosh (^4 ~'") • 7. Find the derivatives of the following, and ...

188 CHAPTER 5 • ELEMENTARY FUNCTIONS Given an elegant argument that explains why t he following functions a.re harmonic. (a} h ...

5.5 • INVERSE TRIGONOMETRIC AND HYPERBOLIC F UNCTIONS 189 We can find the derivatives of any branch of these functions by using ...

190 CHAPTER 5 • ELEMENTARY FUNCTIONS 10 x 10 -3 --0 ..... - 2 2 4 6 8 10 Figure 5.10 A rectangular grid is mapped onto a spider ...

5.5 • INVERSE TRICONOMETRJC AND HYPERBOLIC FUNCTIONS 191 and the corresponding value of the derivative is given by 1. f I ( vz) ...

192 CHAPTER 5 • ELEMENTARY FUNCTI ONS -------.-EXERCISES FOR SECTION 5.5 Find all values of the following. (a) arcsin ~· (b) a ...

cb-~Rter 6 integration Overview Of the two main topics studied in calculus-differentiation and integration- we have so fur only ...

194 CHAPTER 6 • COMPLEX INTEGRATION We generally evaluate integrals of this type by finding the antiderivatives of u and v and e ...

6.1 • COMPLEX INTEGRALS 19 5 We can evaluate each of the integrals via integration by parts. For example, ) t=lf (~ et cos tdt = ...

196 CHAPTER. 6 • COMPLEX INTEGRATION If the limits of integration a.re reversed, then 1b J(t)dt = -1(). f (t)dt. (6-6) The in ...

6.1 • COMPLEX INTEGRALS 197 EXAMPLE 6.4 Use Equation (6-8) to show that ('i. 1 ( • ) i ( • ) lo exp(t+it)dt=z e'- 1 + 2 e'+l. ...

198 CHAPTER 6 • COMPLEX INTEGRATION Show that Jo"" e-•'dt = ~ provided Re (z) > O. Establish the following identities. (a) ...

t(b) ~ (a) A curve that is simple. z(b) z~ (c) A curve that is not simple and not closed. 6.2 • CONTOURS AND CONTOUR INTEGRALS ...

200 CHAPTER 6 • COMPLEX lNTECRATION If C is a smooth curve, then ds, the differential of arc length, is given by ds = V[x' (t)]^ ...

6.2 • CONTOURS AND CONTOUR I NTEGRALS 201 y I + i I + i Figure 6 .4 The polygonal path C = C1 + Cz + C3 from -1 + i to 3 -i. S ...

202 CHAPTER 6 • COMPLEX INTEGRATION y 2 Figure 6.6 Partition and evaluation points for the Riemann sum S (P 8 ). I Definition 6. ...

6.2 • CONTOURS AND CONTOUR INTEGRALS 203 This result compares favorably with the precise value of the integral, which you will s ...