mathematical methods in science and engineering wiley

... 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 fact, the number ... motion and the Fokker-Planck equation. This is an emerging field with enormous potential and with applications to physics, chemistry, biology, engineering, and finance. For beginning researchers ... terest in interdisciplinary studies has brought scientists together from physics, chemistry, biology, economy, and finance and has increased the demand for these courses in which upper-level mathematical

Ngày tải lên: 13/08/2014, 09:21

... that the value of the integral in Equation (3. 38) can be obtained by expanding in powers of t and s and then by comparing the equal powers of tnsm with the left-hand side of Equation ... x and E are defined in terms of the radial distance r and the energy E as x=- r Iri vmw and E= E hW/2 and 1 takes integer values 1 = 0,1,2 i) Examine the nature of the singular point ... 39) and (3. 40) and then the result with ep" to write (3. 41) ORTHOGONALiTY... ORTHOGONALiTY OF LAGUERRE POLYNOMlALS 49 Interchanging the integral and the summation signs and integrating

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 ... 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 ... be defined by a generating function T(x,t) as (6 .47 ) n= 03 90 BESSEL FUNCTIONS 6.3.2 Integral Definitions Bessel function J,(x) also has the following integral definitions: (6 .48 ) and Jn(IL.1

Ngày tải lên: 13/08/2014, 09:21

... exists a minimum value, 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 ... m=O which converges absolutely and uniformly in all subintervals free of points of discontinuity. At the points of discontinuity this series rep resents (as in the Fourier series) the arithmetic ... changes by integer amounts, we... Equation (9.83) and integrating gives us k o ( z )= cacota(z + p ) + sin+ d + p ) ’ (9.89) where p and d are integration constants With these Ico and Icl

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$ ... 178 COORDINATES AND TENSORS which again proves that infinitesimal rotations commute 10.7 CARTESIAN TENSORS Certain physical properties... A(t)r(O) (10.72) Differentiating and using Equation ... 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,

Ngày tải lên: 13/08/2014, 09:21

... polar coordinates (r, 6,Q,) are defined x, as x y z = rsinxsinBcosQ, = rsinXsinBsinq5 = rsinxcos6 w = r cos x 220 COORDINATES AND TENSORS i) Write the line element ds2 = dx2 + dy2 + dz2 + ... be obtained by using dZa a; = - dxp (10. 277 ) In Minkowski spacetime the distance between two infinitesimally close points (events) can be written... Abelian 226 CONTINUOUS GROUPS AND REPRESENTATIONS ... however, the form of the tensor equation remains the same. One of the early uses for tensors in physics was in searching and expressing the coordinate independent properties of crystals. However,

Ngày tải lên: 13/08/2014, 09:21

... the z2-axis, in terms of the components of the angular momentum about the 2-,y-, and z-axes as: L,, = -2- a dB = -sinaL, + cosaL, (11.291) and Inverting these we obtain -sina- d dp cosa- ... sin &+a6 z = rcosf?, and (11.226) 254 CONTINUOUS GROUPS AND REPRESENTATIONS Y’ Fig 11.6 (z,y,z) and the (z‘,y’,z‘) coordinates we obtain (Fig 11.6) L,= (11.227) d icotOsin 484 (11.2 28) (11.229) ... 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

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 ... 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 ... 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

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

... in evaluating definite integrals and finding sums of infinite series In this chapter, we introduce some of the basic... analytic function in and on the closed contour C in a simply ... this integral by first integrating up to an infinitesimally close point, ( a - a), to a and then continue integration on the other side from an arbitrarily close point, ( a + S), to infinity, ... 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

... 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, ... DERIVATIVES 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 ... compared t o a Gaussian APPLICATIONS OF DIFFERINTEGRALS IN SCIENCE AND ENGINEERING 427 Fig 14.7 Probability distribution in random walk and CTRW An important area of application for

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

Ngày tải lên: 13/08/2014, 09:21

... 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 ... where y is real and s is now a complex variable The 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 ... 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

Ngày tải lên: 13/08/2014, 09:21

... applications not just in physics and engineering but also in finance and economics. In applications we frequently encounter caees where a physical property is represented by an intwal, the extremum ... 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 ... time 17.12 In an expanding flat universe the... have no choice but to use integral equations In this chapter we discuss the basic properties of linear integral equations and introduce some

Ngày tải lên: 13/08/2014, 09:21

... 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

... 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 ... dependence is carried in the expansion coefficients Am(r) Operating on @(T,T) H and remembering that H is independent of 7, with we obtain m I m Using Equation (19.250) and the time derivative ... 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

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

Ngày tải lên: 13/08/2014, 09:21

... substances, including type I and III interferons, lactoferrin, β-defensins, and nitric oxide, and also in the production of cytokines and chemokines, which recruit inflammatory cells and influence ... into upper (URTIs) and lower (LRTIs) respiratory infections, depending on the affected organs (noses, sinuses, middle ear, larynx, and pharynx in the URTIs and trachea, bronchi, and lungs in ... g/m3). An interquartile range increase in PM 10 concentration (28.26 g/m3) led to increases in URTI ER visits. The effect was acute, starting at the same day of exposure (lag0) and remaining for

Ngày tải lên: 09/08/2014, 16:21

... MATLAB IN SCIENCE AND ENGINEERING Edited by Tadeusz Michałowski Applications of MATLAB in Science and Engineering Edited by Tadeusz Michałowski Published by InTech ... areas include mathematics, physics, chemistry and chemical engineering, mechanical engineering, biological and medical sciences, communication and control systems, digital signal, image and video ... libraries, and incorporates a high-level programming language. Nowadays, the MATLAB/Simulink package is the world’s leading mathematical computing software for engineers and scientists in industry and

Ngày tải lên: 09/08/2014, 16:21

... learning Supervised learning, also called classification, takes as input a grouping of objects of a so-called “training set” Such a training set... the staining intensity of a silver staining, ... MATLAB in Science and Engineering From Discrete to Continuous Gene Regulation Models – A Tutorial Using the Odefy Toolbox 17 Visualize(r.x,r.y,switchAND.species); resulting in the following trajectories, ... without running short of 64 Applications of MATLAB in Science and Engineering licenses Both licensed MATLAB and MATLAB compiler are available for both Windows and Unix However,

Ngày tải lên: 09/08/2014, 16:21