advanced mathematical methods in science and engineering

... differential equations and evaluating some definite integrals 3 Infinite series, infinite products, asymptotic solutions, and stability cal- culations are other areas, in which complex techniques ... complex techniques are very helpful in certain problems of physics and engineering, which are essentially problems defined in the real domain, complex numbers in quantum mechanics appear as an ... 20292 CONTINUOUS GROUPS AND REPRESEN TATlONS 11.18 Using induction, show that Trang 21of Laplace equation in two dimensions 2 The method of analytic continuation is very useful in finding solutions

Ngày tải lên: 13/08/2014, 09:21

... field lines inside... CLASSIFICATION OF SINGULAR POINTS 13.4 347 CLASSIFICATION OF SINGULAR POINTS Using Laurent the series we can classify singular points of a function Definition I Isolated singular ... < 0, a, = 0 and a-lrnl then # 0, is called a singular point of order m Definition I11 Essential singular point: If m is infinity, then singular point a is called an essential Definition IV Simple ... lines u = c1 and u = c2 (12.125) in the w-plane (Fig. 12.14). The problem is now reduced to finding the equipotentials and the electric field lines between two infinitely long MAPPINGS

Ngày tải lên: 13/08/2014, 09:21

... evaluating definite integrals and finding sums of infinite series In this chapter, we introduce some of the basic properties of fractional calculus along with some mathematical techniques and their ... a and then continue integration on the other side from an arbitrarily close point, (a + S), to infinity, that is, define the integral I as Trang 14t' fig 13.18 Contour G for the Cauchy principal ... u is inside our contour Using the residue theorem we obtain (13.185) In this case we again have two choices for the detour around the singular point on the real axis Again the Cauchy principal

Ngày tải lên: 13/08/2014, 09:21

... (14.97) and choose the integer n as q - n < 0 We now evaluate the differintegral inside the square brackets using formula (14.71) as Combining this with the results in Equations (14.96) and (14.98) ... interpret the left-handed definition] in general the boundary or the initial conditions determine which definition is t o be used It is also possible to give a left-handed version of the Griinwald ... an integer satisfying O < Q < m < Q + 1 (14.161) For the j values that make ( p - Q + j + 1) positive, the gamma function in the denominator is finite, and the corresponding terms in

Ngày tải lên: 13/08/2014, 09:21

... kinetic energy: T h e binomial formula is probably one of the most widely used formulas in science and engineering An important application of the binomial formula was given by Einstein... ... INFINITE PRODUCTS Infinite products are closely related to infinite series Most of the known functions can be written as infinite products, which are also useful in calculating some of... INFINITE ... f(0)l-... be seen by taking Mi = /ail Sz in the M-test 15.8.2 Continuity In a power series, since every term, that is, u,(x) = unlcn, is a continuous function and since in the interval -S 5 2

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... function in the infinite interval (-CO,CO) We now consider a nonperiodic function in the infinite interval (-o,co) Physically this corresponds to expressing an arbitrary signal in terms of sine and ... transform is defined as and it is usually encountered in potential energy calculations in cylindrical coordinates Another useful integral transform is the Mellin transform: (16.14) The Mellin transform ... transform is defined as ( 16.12) Trang 7DERIVATION OF THE FOURIER INTEGRAL 479 and it is very useful in finding solutions of systems of ordinary differential equations by converting them into a system

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... variational problems encountered in physics and engineering are expressed in terms of an integral: (17.1) where y(x) is the desired function and f is a known function depending on y, its derivative ... in engineering problems we encounter functionals given as (17.66) where y(") stands for the nth order derivative, the independent variable x takes values in the closed interval [a,b], and ... equation of a straight line, where the integration constants a and b are to be determined from the coordinates of the end points The shortest paths between two points in a given geometry are

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... Xj and take its complex conjugate as (18.97) Multiplying Equation (18.96) by Xjy;(z) and Equation (18.97) by Xiyi(z), and integrating over x in the interval [a, b] we obtain two ... chapter, we introduce the basic features of both the time-dependent and the timeindependent Green’s functions, which have found a wide range of applications in science and engineering. .. ... tool in solving differential equations They are also very useful in transforming differential equations into integral equations, which are preferred in certain cases like... 19.108) Using

Ngày tải lên: 13/08/2014, 09:21

... However, in electrostatics we usually deal with cases in which we are interested in finding the potential of a charge distribution in the presence Trang 12G(?,?‘) and Equation (19.234) with ẳ), and ... surface enclosing the volume V with the outward unit normal ii, we obtain Interchanging the primed and the unprimed variables and assuming that the Green’s function is symmetric in anticipation ... Using this in Equation (19.271) we obtain the propagator, G1(7'),7')", T ~ , T o ) , that takes us from TO to 7 2 in a single step in terms of the propagators, that take us from 70 to 7 1 and

Ngày tải lên: 13/08/2014, 09:21

... also leads to many interesting applications in field theory. In this Chapter we introduce the basic features of this technique, which has many interesting existing applications and tremendous potential ... FUNCTIONS AND PATH INTEGRALS 20.7.2 Schrodinger Equation in the Presence of Interactions In the presence of interactions the Schrdinger equation is given as a*(x,t) ~- at Making the transformation... ... t] denotes all continuous paths starting from (20,t o ) and ending a t ( 2 ,t) Before we discuss techniques of evaluating path integrals, we METHODS OF CALCULATING PATH INTEGRALS 647 should

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... Cosine and Sine Integrals 649 13.7 Contour Integration and Branch Cuts 652 13.8 Exploiting Symmetry 655 13.8.1 Wedge Contours 655 13.8.2 Box Contours 658 13.9 Definite Integrals Involving ... Piecewise Continuous Coefficients and Inhomogeneities 1071 21.5 Inhomogeneous Boundary Conditions 1074 21.5.1 Eliminating Inhomogeneous Boundary Conditions 1074 21.5.2 Separating Inhomogeneous ... 239 7.2 The Point at Infinity and the Stereographic Projection 242 7.3 A Gentle Introduction to Branch Points 246 7.4 Cartesian and Modulus-Argument Form 246 7.5 Graphing Functions of

Ngày tải lên: 06/08/2014, 01:21

... the function is defined. Hint 1.4 Find the slope and x-intercept of the line. Hint 1.5 The inverse of the function is the reflection of the function across the line y = x. Hint 1.6 The formu la for ... degrees Celsius and f denote degrees Fahrenheit. The line passes through the points (f, c) = (32, 0) and (f, c) = (212, 100). The x-intercept is f = 32. We calculate the slope of the line. slope = ... quantity having both a magnitude and a direction. Examples of vector quantities are velocity, force and position. One can represent a vector in n-dimensi onal space with an arrow whose initial point

Ngày tải lên: 06/08/2014, 01:21

... Removable discontinuity, a Jump Discontinuity and an Infinite DiscontinuityBoundedness A function which is continuous on a closed interval is bounded in that closed interval Nonzero in a Neighborhood ... continuous for each point x ∈ (a, b) andlimx→a+y(x) = y(a) and limx→b−y(x) = y(b).Discontinuous Functions If a function is not continuous at a point it is called discontinuous at that point Iflimx→ξy(x) ... Trang 2defined, limx→ξy(x) exists and limx→ξy(x) = y(ξ) A function is continuous if it is continuous at each point in itsdomain A function is continuous on the closed interval [a, b] if

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... bounded, there is a continuous, increasing function (δ) satisfying, e(x, δ) ≤ (δ),for all x in the closed interval Since (δ) is continuous and increasing, it has an inverse δ() Now note that ... expression to obtain anindeterminate of the form ∞∞ and then apply L’Hospital’s rule lim x→0 ln(sin x)1/ sin x = limx→0 Trang 24Chapter 4Integral Calculus 4.1 The Indefinite Integral The opposite ... the indefinite integral The indefinite integral of a function f (x) Z f (x) dx = f (x) While a function f (x) has a unique derivative if it is differentiable, it has an infinite number of indefinite

Ngày tải lên: 06/08/2014, 01:21

... (mathematica/calculus/integral/improper.nb)EvaluateR04 (x−1)1 2 dx Hint, Solution Exercise 4.18 (mathematica/calculus/integral/improper.nb)EvaluateR1 Hint, Solution Trang 7Hint 4.9CONTINUE Hint 4.10 a Evaluate the integral b Use integration by ... Trang 10Solution 4.4Zcos xsin x dx = Z1sin x x3− 5 13 = 12 Trang 11Solution 4.7Let u = sin x and dv = sin x dx Then du = cos x dx and v = − cos x Z π 0 sin2x dx = − sin x cos xπ0 + Z π 0 cos2x ... interval (a b) We define ∆xi = xi+1− xi and ∆x = maxi∆xi and choose ξi ∈ [xi xi+1] Z b a √ x dx = Z 1 0 √ x dx + Z 2 0 + 2 3x 3/2 2 0 ddx Trang 20Solution 4.4First we expand the integrand in

Ngày tải lên: 06/08/2014, 01:21

... 5 cos θsin(5θ) = 5 1 − sin2θ
2sin θ − 10 1 − sin2θ
sin3θ + sin5θ sin(5θ) = 16 sin5θ − 20 sin3θ + 5 sin θ Addition We can represent the complex number z = x + ıy = r eıθ as a vector in Cartesian ... sin2θ − ı5 3 cos2θ sin3θ +54 cos θ sin4θ + ı5 5 sin5θ = cos5θ − 10 cos3θ sin2θ + 5 cos θ sin4θ
+ ı 5 cos4θ sin θ − 10 cos2θ sin3θ + sin5θ
Then we equate the real and imaginary parts cos(5θ) ... Sincep1 + 1/x2 > 1/x, the integral diverges The length is infinite We find the area of S by integrating the length of circles A = Z ∞ 1 2π x dxThis integral also diverges The area is infinite

Ngày tải lên: 06/08/2014, 01:21

... purposes Instead, we introduce complex infinity or the point at infinity as the limit of going infinitely far alongany direction in the complex plane The complex plane together with the point at infinity ... line Thus signed infinity makes sense By going up or down we respectively approach +∞ and −∞ In thecomplex plane there are an infinite number of ways to approach infinity We stand at the origin, ... curve enclosing the point at infinity However, in the stereographic projection, the pointat infinity is just an ordinary point (namely the north pole of the sphere) In this section we will introduce

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... cos(z) + ı sin(z) + cos(z) − ı sin(z) 2 = cos z e ız − e −ız ı2 = cos(z) + ı sin(z) − cos(−z) − ı sin(−z) 2 = cos(z) + ı sin(z) − cos(z) + ı sin(z) 2 = sin z We separate the sine and cosine into their ... cosine into their real and imaginary parts. cos z = cos x cosh y − ı sin x sinh y sin z = sin x cosh y + ı cos x sinh y For fixed y, the sine and cosine are oscillatory in x. The amplitude of ... with increasing |y|. See Figure 7.18 and Figure 7.19 for plots of the real and imaginary parts of the cosine and sine, respectively. Figure 7.20 shows the modulus of the cosine and the sine.

Ngày tải lên: 06/08/2014, 01:21

... branch points in the finite plane to the branch point at infinity and a branch cut connecting the remaining two branch points See Figure 7.51bcd Note that in walkingaround any one of the finite ... Reverse the orientation of thecontour so that it contains the point at infinity and does not contain any branch points in the finite complex plane.Hint 7.29 Factor the polynomial The argument of z1/4 ... + ı cos x)
= sin x cosh y + ı cos x sinh y sin z = qsin2x cosh2y + cos2x sinh2y exp(ı arctan(sin x cosh y, cos x sinh y)) = qcosh2y − cos2x exp(ı arctan(sin x cosh y, cos x sinh y)) = r1 2(cosh(2y)

Ngày tải lên: 06/08/2014, 01:21

... branch points in the finite plane to the branch point at infinity and a branch cut connecting the remainingtwo branch points See Figure 7.53bcd Note that in walking around any one of the finite ... branch point at infinity.The first two sets of branch cuts in Figure 7.33do not permit us to walk around any of the branch points, includingthe point at infinity, and thus make the function single-valued ... be a branchpoint If f (z) has only a single branch point in the finite complex plane then it must have a branch point at infinity If f (z) has two or more branch points in the finite complex

Ngày tải lên: 06/08/2014, 01:21