mathematical methods in science polya pdf

... an integral part of the learning process. In a vast area like mathematical methods in science and engineering, there is always room for new approaches, new applications, and new topics. In ... complex inte- grals, series, and analytic continuation along with the methods of evaluating some definite integrals. Chapter 14 introduces the basics of fractional calculus. After introducing ... Comparing this with... l in the generating function Equation (2. 65) we find (2. 85) Expanding the left-hand side by using the binomial formula and comparing equal powers o f t , we obtain 9

Ngày tải lên: 13/08/2014, 09:21

... operator in cylindrical coordinates 10.24 In four dimensions spherical polar coordinates (r, 6,Q,) are defined x, as x y z = rsinxsinBcosQ, = rsinXsinBsinq5 = rsinxcos6 w = r cos x 220 COORDINATES ... general linear transformation [Eq (10.250)] matrix elements a$ can be obtained by using dZa a; = - dxp (10. 277 ) In Minkowski spacetime the distance between two infinitesimally close points (events) ... is called pseudo-Euclidean. In this line element c is the speed of light representing the maximum velocity in nature. An interesting property of the Minkowski spacetime is that two

Ngày tải lên: 13/08/2014, 09:21

... 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 ... 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... ... 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

Ngày tải lên: 13/08/2014, 09:21

... language in the socio-cultural approach is considered extremely relevant in socially negotiating and meaning making The widening interest in situated learning resides in the belief that learning is ... illegally dumping and, in turn, poisoning the residents in the area As she digs deeper, Erin finds herself a leading figure in a series of events that would involve her law firm in one of the ... mediating variables that determine the effectiveness of cinema for enhancing science teaching and learning Science educators have argued that science classrooms ought to be active learning environments

Ngày tải lên: 23/03/2014, 11:21

... method, find a n infinite series solution about x = 0 in the interval [0,m] Check t h e convergence of your solution Should your solution be finite everywhere, including the end points of your interval? ... finite solution in the interval [O,m],we terminate the series [Eq (3. 10)]by restricting po (energy) to the values po = 2 ( N + 1 + l), N = 0, 1,2, (3. 13) Since 1 takes integer values, we introduce ... P;"(z), m 2 0. 2.13 in the interval [-1,1]. 2.14 A metal sphere is cut into sections that are separated by a very thin insulating material. One section extending from 6 = 0 to

Ngày tải lên: 13/08/2014, 09:21

... hypergeometric functions is: This integral, which can be proven by expanding (1- tz)- in binomial series and integrating term by term, transforms into an integral of the same type by Euler’s ... (x)= The main reason for our interest in hypergeometric functions is that so many of the second-order linear ordinary differential equations encountered in physics and engineering ... FUNCTIONS Using the series definitions of the Bessel functions we can obtain the following recursion relations and First by adding and then by subtracting these equations we also obtain the relations

Ngày tải lên: 13/08/2014, 09:21

... mmin, thus determining X as X = mmin (9.143) To find mminwe equate the two expressions [Eqs (9.141) and (9.143)] for X t o obtain (9.144) mmin -1 = (9.146) Since m changes by integer amounts, ... is square integrable means that the integral of the square of the derivative is finite for all the subintervals of the fundamental domain [a, b] in which the function is continuous. 8.5 ... eigenvalues. In other words, {urn(.)} spans the infinite dimensional vector space of square integrable functions. The inner product (analog of dot product) in Hilbert space is defined as (8.59)

Ngày tải lên: 13/08/2014, 09:21

... cos+cos - cos8 sin +sin cos - cos 6sin +cos sin 8 sin - sin + (10. 56) (10.57) ++ + + cos+ sin cos8cos+sin sin cos 6cos +cos -sin 8cos q5 - sin + ++ The inverse of... -sin4 1 - sin $ sin4cos$ cos~cos$ ... COORDINATES In threedimensional Euclidean space a Cartesian coordinate system can be constructed by choosing three mutually orthogonal straight lines A point is defined by giving its coordinates, ... sin4cos$ cos~cos$ Reversing the order we get R2R1 = [ (10.93) - sin $ cos Cp cos $ sin $ sin Cp 0 cos Cp sin 4 sin $ - cos sin q5 cos $ cos 4 (10.94) $ J It is clear that for finite rotations these

Ngày tải lên: 13/08/2014, 09:21

... cosa sin@ay... entirely in terms of the original coordinate axis For this we first need to find how the operator R transforms under coordinate transformations We now transform to a new coordinate ... the yn, matrix of this coordinate transformation with the letter R We are interested in expressing the operator R in terms of the ( z ~ , zn) coordinates... Coordinates One of the disadvantages ... (11.229) L*... (Ly)n obtained in Section 11.11.7, we write this as dm,,(P) 1 = Jmmg - i ( L g ) m m , =[: i I;f 0 0 :I isinP[ 1 1/2 -l/2 i 4 0 -1/2 0 1/2 Finally adding these we find 0 d k l m (PI

Ngày tải lên: 13/08/2014, 09:21

... evaluating some definite integrals 3 Infinite series, infinite products,... (x,y) in the z-plane to another point (u, in the w-plane, v) which implies that curves and domains in the z-plane ... useful in finding solutions of Laplace equation in two dimensions 2 The method of analytic continuation is very useful in finding solutions of differential equations and evaluating some ... the point is mapped to w = fie'" = -6 (12 .99 ) in the w-plane However the coordinates (12 .97 ) and (12 .98 ) represent the same point in the z-plane In other words, a single point in

Ngày tải lên: 13/08/2014, 09:21

... 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 Pole: In Definition 11, if ... 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 ... point: If a function is not analytic at a but analytic at every other point in some neighborhood of a, then is called an isolated singular point Definition I1 Essential singular point: In

Ngày tải lên: 13/08/2014, 09:21

... some useful mathematical techniques in evaluating definite integrals and finding sums of infinite series In this chapter, we introduce some of the basic... analytic function in and on ... continue integration on the other side from an arbitrarily close point, ( a + S), to infinity, that is, define the integral I as (13.175) This is called taking the Cauchy principal value of the integral, ... Hint First define in - -Ji fin1, 378 COMPLEX INTEGRALS AND SERIES Using this result, define hil)(z) a contour integral in the complex j’-plane as (j’= t’ z ’ ,where s) + d I d dt xdx‘ Indicate

Ngày tải lên: 13/08/2014, 09:21

... Writing the remaining part of the $, dz integral we get (:!$I (14.82) After taking the limit we substitute this into the definition [Eq. (14.74)] to obtain (14.83) Simplifying this ... 14.6.5 Evaluating Definite Integrals by Differintegrals We have seen how analytic continuation and complex integral... solution involves two integration constants and a divergent integral However, ... AND INTEGRALS: "DIFFERINTEGRALS" where k is again an integer satisfying Equation (14.179) Even though for dynamic processes it is difficult t o interpret the left-handed definition] in general

Ngày tải lên: 13/08/2014, 09:21

... FOURIER INTEGRAL 16.2.1 Fourier Series Fourier series are very useful in representing a function in a finite interval, like [0,27r] or [-L, L], or a periodic function in the infinite ... 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 cosine ... contour for the above integral is an infinite straight line passing through the point y and parallel t o the imaginary axis in the complex s-plane y is chosen such that all the singularities of

Ngày tải lên: 13/08/2014, 09:21

... problem: Find the shape of the curve joining two points, along which a particle, initially a t rest, falls freely under the influence of gravity from the higher point to the lower point in the ... CLASSIFICATION OF INTEGRAL EQUATIONS Linear integral equations are classified... Volterra equations also have the following kin& a#O kind1 =1 kind I1 cr = a(%) kind I11 cy When F ( z ) is zero, the integral ... homogeneous Integral equations can also be defined in higher ciimen%ions In two dimensions we can write a linear integral equation as 18.2 INTEGRAL AND DIFFERENTIAL EQUATIONS Some integral

Ngày tải lên: 13/08/2014, 09:21

... to introduce the Hilbert-Schmidt theory, where an eigenvalue problem is defined in terms of linear integral operators. 18.6.1 Using the Fredholm equation of the second kind, we can define ... very useful in transforming differential equations into integral equations, which are preferred in certain cases like... 19.108) Using this Green’s function, solution of the inhomogeneous ... time-dependent and the timeindependent Green’s functions, which have found a wide range of applications in science and engineering. .. solution exists and then find it 18.11 1 Using the Neumann

Ngày tải lên: 13/08/2014, 09:21

... (19.217) This is also the integral equation version of the time-independent Shrdinger equation for bound states. CaseII: X>O: In this case the denominator in the definition [Eq. (19.197)] ... well defined everywhere on the real k-axis. Taking the inverse Fourier transform of this we get We can now evaluate this integral in the complex k-plane using the complex contour integral ... 2 in a single step in. .. obtaining the Green’s function as G(z,z’) = CYI (z) + Dy2(~), J: < z’, where W[yl(z),92(z)]is the Wronskian defined as W[91,y2] = ylya - y2y/i Now impose the single-point

Ngày tải lên: 13/08/2014, 09:21

... clearer, but 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 ... 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... ... l, x... restrictions To bring this integral into a form that can be evaluated in practice, we introduce the phase space lattice by dividing the time interval t E [t’, into t”] N 1 slices as +

Ngày tải lên: 13/08/2014, 09:21

... MATHEMATICAL METHODS IN SCIENCE AND ENGINEERING MATHEMATICS AND MIND 5 1.3 MATHEMATICS AND MIND Almost, everywhere mathematics is a very useful and powerful language in expressing ... an integral part of the learning process. In a vast area like mathematical methods in science and engineering, there is always room for new approaches, new applications, and new topics. In ... naturally comes from the inner efficiency of our brain. Research on subjects like brain stimulators, hard wiring of our brain, and mind reading machines are all aiming at a faster and much...

Ngày tải lên: 22/03/2014, 13:20

... Sage Publications. Harding, S. 1987. Introduction: Is there a feminist method? In Sandra Harding (ed.), Feminism and Methodology: Social Science Issues. Bloomington, Indiana: Indiana University Press. Kuhn, ... Research Methods in Public Administration AU5384_C002 Final Proof page 14 19.9.2007 4:33pm Compositor Name: BMani 14 Handbook of Research Methods in Public Administration 1 The Logic of Inquiry in ... imprint of the Taylor & Francis Group, an informa business Boca Raton London New York Handbook of Research Methods in Public Administration Miller/Handbook of Research Methods in Public Administration...

Ngày tải lên: 15/03/2014, 05:20