Begin2.DVI

Figure 12-10. Representing function of complex variable as mapping. Polar coordinates (r, θ)in the z−plane correspond to polar c ...

Example 12-17. (Representing function of a complex variable as mapping) Consider the complex function ω=f(z) = z^2 = (x+i y)^2 = ...

495 z-plane Mapping ω-plane 1. ω = z 2 2. ω = z 2 3. ω = z^2 4. ω = z 2 Figure 12-11. Selected images illustrating the mapping ω ...

Derivative of a Complex Function The derivative of a complex function ω=f(z) = u(x, y ) + i v(x, y ), where z=x+i y , is defined ...

If the limit in equation (12.178) approaches zero along the path 2 of figure 12-12, then the definition given by equation (12.17 ...

Contour integration Let C denote a curve in the z-plane connecting two points z=aand z=bas illustrated in figure 12-13. Figure 1 ...

in equation (12.183) is called the complex line integral of f(z)along the curve Cand is denoted ∫ C f(z)dz = limn→∞ n∑− 1 i=0 f( ...

length Lbetween two points a= [x(ta), y (ta)] and b= [x(tb), y (tb)] on the curve is given by the integral L= ∫ C ds = ∫tb ta √( ...

C =C 1 ∪C 2 ∪···∪ Cm, then the line integral can be broken up and written as a summation of the line integrals over each section ...

Table 12.1 Short Table of Integrals 1. ∫ zndz =z n+1 n+ 1 +c, n
=− 1 11. ∫ sinh z dz = cosh z+c 2. ∫ dz z = log z+c 12. ∫ cosh ...

2. If f(t)is continuous and ta< tc< tb, then ∫tb ta f(t)dt = ∫tc ta f(t)dt + ∫tb tc f(t)dt 3. The modulus of the integral ...

and consequently, ∫ C f(z)dz = ∫ C (u+iv )(dx +idy ) = ∫ C (u dx −v dy ) + i ∫ C (v dx +u dy) = ∫t 2 t 1 ∂U ∂x x′(t)dt +∂U ∂y y′ ...

The arrow on the circle indicating the direction of integration as being clockwise or counterclockwise as viewed looking down on ...

by |z|= √ x^2 +y^2 which represents the distance of the point zfrom the origin in the z−plane. (ii) The quantity |z−z 0 |=Rrepre ...

R 2 are positive constants with R 2 < R 1. The annular region of convergence is the intersection of these two regions. Figure ...

will be of great assistance in dealing with Laurent series. Example 12-21. (Laurent series) Express the function f(z) = z (z−1)( ...

illustrated in the figure 12-16. (b) If one had used the binomial expansion on the last term in the representation (i) for f(z), ...

APPENDIX A Units of Measurement The following units, abbreviations and prefixes are from the Syst`eme International d’Unit`es (d ...

DERIVED UNITS Name Units Symbol Area square meter m^2 Volume cubic meter m^3 Frequency hertz Hz (s−^1 ) Density kilogram per cub ...

APPENDIX B Background Material Geometry Rectangle Area=(base)(height)=bh Perimeter= 2b+ 2h Right Triangle Area= 1 2 (base)(heigh ...