Mathematics of Physics and Engineering
186 Algebra of Complex Numbers there exists a neighborhood of the point that lies entirely in the set. FOR EXAMPLE, the center o ...
AC Circuits 187 4.1.3 Applications to Analysis of AC Circuits In this section, we continue to use i as the notation for the imag ...
188 Algebra of Complex Numbers "VL E = VR + Vc + VL Vc •+J •— VR I Series L Ic IR E •+T *— II I = IR + IC + IL Parallel Fig. 4.1 ...
AC Circuits 189 the resonance frequency wo = l/VLC. Now let us consider the parallel RCL circuit on the right-hand side of Figur ...
190 Functions of a Complex Variable 4.2 Functions of a Complex Variable 4.2.1 Continuity and Differentiability The study of func ...
Cauchy-Riemann Equations 191 say that a function is analytic if it is analytic somewhere. An entire function is a function that ...
192 Functions of a Complex Variable e(z) = E(x,y) + ie2(x,y), f'(zo) = A + iB, r = \z\, we find that, for all x, y sufficiently ...
Cauchy-Riemann Equations 193 and v have continuous partial derivatives of every order. Recall that the alternative notation for ...
194 Functions of a Complex Variable imply ur = r~^1 ve, vr = -r~^1 ue, (4.2.3) and conversely, (4-2.3) imply (4-2.2). (b) Verify ...
The Theorem and Formula of Cauchy 195 gral over a closed curve C (that is, a curve for which r(a) = r(b)) is often denoted by §c ...
(^196) Functions of a Complex Variable of |/(.z)| on the curve C. The result is the following inequality for the integral (4.2.4 ...
The Theorem and Formula of Cauchy 197 dence property in a simply connected domain has an anti- derivative there. Theorem 4.2.2 A ...
(^198) Functions of a Complex Variable the best proof, because the definition of an analytic function requires only the existenc ...
The Theorem and Formula of Cauchy 199 Proof. Remember that the change of the orientation reverses the sign of the line integral; ...
(^200) Functions of a Complex Variable To complete the proof of (4.2.11), it is therefore enough to show that lira d> /(C) - ...
The Theorem and Formula of Cauchy 201 n-th order derivative /(") of / exists at every point in G and as long as the closed disk ...
202 Functions of a Complex Variable root. Hint: if the polynomial P = P(z) has no roots, then l/P(z) is a bounded entire functio ...
Conformal Mappings 203 corresponding to some level sets of u and v. Show that, at every point of intersection, the angle between ...
204 Functions of a Complex Variable EXERCISE 4.2.24. B Using the Cauchy-Riemann equations (4-2.2), verify that the Jacobian of t ...
Conformal Mappings 205 real part of w. Since the collection of all w with the same imaginary part is a line parallel to the real ...
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