By Harold Cohen
Complex research with functions in technology and Engineering
Complex research with functions in technology and Engineeringweaves jointly conception and large applications in arithmetic, physics and engineering. during this variation there are numerous new difficulties, revised sections, and a completely new bankruptcy on analytic continuation. This paintings will function a textbook for undergraduate and graduate scholars within the components famous above.
Key beneficial properties of this moment Edition:
Excellent assurance of issues akin to sequence, residues and the overview of integrals, multivalued capabilities, conformal mapping, dispersion family members and analytic continuation
Systematic and transparent presentation with many diagrams to explain dialogue of the material
Numerous labored examples and a great number of assigned problems
Read or Download Complex Analysis with Applications in Science and Engineering PDF
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Additional info for Complex Analysis with Applications in Science and Engineering
These are the two points on the unit circle shown in fig. 4a. 71c) respectively. 71d) is the same as the root obtained for k = 0. The three distinct roots of 1 are the points on the circumference of the unit circle shown in fig. 4b. 72) with 0 ≤ k ≤ (N – 1) or some equivalent range of k. Just as with the roots of +1, the roots of –1 for k ≥ N are the same as those for 0 ≤ k ≤ (N – 1). We see from eq. 72 that the roots of –1 are also of modulus 1 and so represent points on the unit circle. 74c) which is the same as the root for k = 0.
53b) Exponential representations of trigonometric functions Setting r = 1, eqs. 55 are referred to as the exponential representations of cosθ and sinθ. The exponential representations of tanθ, cotθ, secθ, and cscθ are obtained straightforwardly from these expressions. 56) Hyperbolic functions The hyperbolic functions are defined from the trigonometric functions for imaginary angles. 57) with w real. Then eqs. 59b) cos(iw) = and sin(iw) = from which sinh w = The real functions, cosh w and sinh w, are the hyperbolic cosine and hyperbolic sine of w.
28) which is referred to as the polar or exponential form. 3a) in the three standard forms, we first write 1 + i in polar form. 2 Cartesian, Trigonometric, and Polar Forms 15 The Cartesian form of this complex number is obtained from eq. 47) Expanding the left side of eq. 47, we obtain deMoivre’s theorem n Â (i) m m=0 ( )( ) n! (n - m)! 48) Then È n ˘ n! (n - m)! 49a) È n ˘ n! (n - m)! 50b) Therefore, cos(nθ ) = n Â (-1) m2 m=0 m even n! (n - m)! 51a) and sin(nθ ) = n Â (-1) m =1 m odd ( m -1) / 2 n!