MATHEMATICAL METHODS for Scientists and Engineers
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1 Chapter 18 Figures From MATHEMATICAL METHODS for Scientists and Engineers Donald A. McQuarrie
2 For the Novice Acrobat User or the Forgetful When ou opened this file ou should have seen a slightl modified cover of the book Mathematical Methods for Scientists and Engineers b Donald A. McQuarrie, a menu bar at the top, some inde markers at the left hand margin, and a scroll bar at the right margin. Select the View menu item in the top menu and be sure Fit in Window and Single Page are selected. elect the Window menu item and be sure Bookmarks and Thumbnails ARE NOT selected. You can and probabl should make the top menu bar disappear b pressing the function ke F9. Pressing his ke (F9) again just toggles the menu bar back on. You ma see another tool bar that is controlled b unction ke F8. Press function ke F8 until the tool bar disappears. In the upper right hand corner margin of the window containing this tet ou should see a few small boes. DO NOT move our mouse to the bo on the etreme right and click in it; our window will disappear! Move our mouse to the second bo from the right and click (or left click); the window containing this tet should enlarge to fill the screen. Clicking again in this bo will shrink the window; clicking again will return the displa to full screen. The prefered means of navigation to an desired figure is controlled b the scroll bar in the column at the etreme right of the screen image. Move our mouse over the scroll bar slider, click, and hold the mouse button down. A small window will appear with the tet "README (2 of 37)". Continuing to hold down the mouse button and dragging the button down will change the tet in the small window to something like "18.4 (6 of 37)". Releasing the mouse button at this point moves ou to Figure 18.4 of Chapter 18. The (6 of 37) indicates that Figure 18.4 resides on page 6 of the 37 pages of this document. ANIMATIONS There are no animations in this chapter at this time.
3 Figure 18.1 An illustration of a branch cut along the positive ais for the function f (z) = z 1/2. From MATHEMATICAL METHODS for Scientists and Engineers, Donald A. McQuarrie, Copright 2003 Universit Science Books
4 Figure 18.2 A branch cut along the positive ais for f (z) = ln z and a branch point at the origin.
5 f v z 0 d e w l z u Figure 18.3 An illustration of the limit of a function f (z) in the comple plane. (a) The point z lies within a d neighborhood of z 0. (b) The point w = f (z) is lies within an e neighborhood of l.
6 z 0 Figure 18.4 An illustration that z 0 0 is not a branch point of ln z, but z 0 = 0 is.
7 z a q Figure 18.5 A pictorial aid to Eample 5.
8 Figure 18.6 The families of curves u(, ) = 2-2 = c 1 (dashed) and v(, ) = 2 = c 2 (solid), showing that the are orthogonal. From MATHEMATICAL METHODS for Scientists and Engineers, Donald A. McQuarrie, Copright 2003 Universit Science Books
9 z 1 z 2 z 3 z n - 1 a b Figure 18.7 The subdivision of a path from a to b in the comple plane subdivided into n segments.
10 H+a, +al B H-a, -al A Figure 18.8 Two different paths along which to integrate f (z) = cos z.
11 H+a, +al B O H-a, -al A Figure 18.9 A region in which the two paths in Figure 18.8 lie.
12 HaL HbL Figure An eample of (a) a simple, closed curve and (b) one that is not.
13 Figure An illustration of z traversing in a positive sense around a closed curve. The region bounded b the closed curve alwas lies to the left as z advances. From MATHEMATICAL METHODS for Scientists and Engineers, Donald A. McQuarrie, Copright 2003 Universit Science Books
14 G R Figure An eample of a multipl-connected region. From MATHEMATICAL METHODS for Scientists and Engineers, Donald A. McQuarrie, Copright 2003 Universit Science Books
15 R Figure An indication of the positive direction of the traverset of the boundar of a multiplconnected region. Once again, the region alwas lies to the left as z advances. From MATHEMATICAL METHODS for Scientists and Engineers, Donald A. McQuarrie, Copright 2003 Universit Science Books
16 a C b R Figure An illustration to aid in the proof of Equation 10.
17 C 1 z 0 C 0 Figure Two simple closed curves, C 0 and C 1, in the comple plane surrounding a point z 0.
18 O Figure An arbitrar simple closed curve surrounding the origin.
19 e e 1 2 Figure The deformation of the circle described b z = 3 into two circles of radii e enclosing the points z = 1 and z = 2. From MATHEMATICAL METHODS for Scientists and Engineers, Donald A. McQuarrie, Copright 2003 Universit Science Books
20 C 2 C 2 Æ z 0 C 1 C 0 Figure A cut from C 0 to C 1 and back for the two simple closed curves shown in Figure 18.15, along with an indication of the positive transits of z along C 0 and C 1 and back and forth along C 2. The two transits along C 2 are separated in the figure onl for illustrative purposes.
21 Ha, al H-a, -al Figure The counterclockwise integration path around the square whose vertices are (±a, ±a).
22 R Figure A semicircular contour of radius R in the upper half plane.
23 H1.5, 1.5L H-1, -1L Figure The closed curve used in Problem 17.
24 a Figure A contour C in Equation 1 along with the point a. From MATHEMATICAL METHODS for Scientists and Engineers, Donald A. McQuarrie, Copright 2003 Universit Science Books
25 2 Figure A rectangle surrounding the point z = 2.
26 rom MATHEMATICAL METHODS for Scientists and Engineers, Donald A. McQuarrie, Copright 2003 Universit Science Books 1 C 2 C 1-1 Figure The two contours to be used to evaluate the integral in Eample 2.
27 1 C 2 C C 1 Figure The deformation of the contour described b z = 2 into the contours C 1 and C 2 in order to evaluate the integral given in Eample 4. From MATHEMATICAL METHODS for Scientists and Engineers, Donald A. McQuarrie, Copright 2003 Universit Science Books
28 C 1 R Figure The contour for Problem 15.
29 R z 0 C a z r Figure The geometr used to derive Equation 9 from Equation 5. rom MATHEMATICAL METHODS for Scientists and Engineers, Donald A. McQuarrie, Copright 2003 Universit Science Books
30 Figure The four roots of z = 0.
31 C 4 C 3 C 1 z a z C 2 z C Figure The geometr used to derive Equations 12 through 14. The point z lies in the region between C 1 and C 2. From MATHEMATICAL METHODS for Scientists and Engineers, Donald A. McQuarrie, Copright 2003 Universit Science Books
32 C 2 Figure The contour used in Eample1 and the singular points of the integrand.
33 3 C 4-3 Figure The contour described b z = 4 and the singular points of the integrand of the integral in Eample 1. From MATHEMATICAL METHODS for Scientists and Engineers, Donald A. McQuarrie, Copright 2003 Universit Science Books
34 3 C 1 C 2-3 C 3 rom MATHEMATICAL METHODS for Scientists and Engineers, Donald A. McQuarrie, Copright 2003 Universit Science Books Figure The deformation of the contour in Figure
35 H 4, 8) 2p -2 p Figure The contour used in Problem 21.
36 1 Figure The contour used in Problem 23.
37 H5, 2L p Figure The contour used in Problem 24.
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