MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 18 doc

... 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 ... this method by evaluating the path integral of a functional F[x(T)] = z(T), in the interval [O,t] via the unpinned Wiener measure. Let 7 be any time in the interval...

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 ... (4.52) 'I A This Page Intentionally Left Blank ORTHOGONALiTY OF LAGUERRE POLYNOMlALS 49 Interchanging the integral and the summation signs and integrating with re- spect...

Ngày tải lên: 13/08/2014, 09:21

... DIFFERINTEGRALS IN SCIENCE AND ENGINEERING 427 Fig. 14.7 Probability distribution in random walk and CTRW An important area of application for fractional derivatives is that the ex- traordinary ... dPaex r*(a,z) = r(a)ecX- dx- 14.7 APPLICATIONS OF DIFFERINTEGRALS IN SCIENCE AND ENGINEERING 14.7.1 Continuous Time Random Walk (CTRW) We have seen that the...

Ngày tải lên: 13/08/2014, 09:21

... for the above integral is an infinite straight line passing through the point y and parallel to the imaginary axis in the complex s-plane. y is chosen such that all the singularities of ... 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...

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 ... 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 contou...

Ngày tải lên: 13/08/2014, 09:21

... definition, 429 Cauchy integral formula, 390 Griinwald definition differintegrals, 385 Laplace transforms, 396 484 Fractional derivatives MATHEMATICAL METHODS IN SCIENCE AND ENGINEERING ... 1 78 1 78 179 180 181 183 164 186 188 188 188 189 189 189 189 190 193 195 196 197 197 197 199 801 201 x CONTENTS 10.10.5 Four- Tensors in Minkowski Spacet...

Ngày tải lên: 13/08/2014, 09:21

... points x = fl and z = 0. If we write x = fl in the generating function Equation (2.65) we find (2.85) Expanding the left-hand side by using the binomial formula and comparing ... 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. I...

Ngày tải lên: 13/08/2014, 09:21

... equations in physics and engineering can be solved by the method of separation of variables. This method helps us to reduce a second-order partial differential equation into a set of ordinary ... second kind. Note that N,(z) and Jm(z) are linearly independent even for the integer values of m. Hence it is common practice to take N,(z) and Jm(z) as the two lin...

Ngày tải lên: 13/08/2014, 09:21

... Cancelling m on both sides and noting that (9.166) (9.167) and using Equation (9.164) we finally write Similarly We now define the ladder operators L+ and L- for the m index of ... (9.141) and (9.143)] for X to obtain (9.144) mmin = -1. (9.146) Since m changes by integer amounts, we could write hin = mmax - integer -1 = 1 - integer 21 = integer....

Ngày tải lên: 13/08/2014, 09:21

... choosing three mutually orthogonal straight lines. A point is defined by giving its coordinates, (q,z2,q), or by using the position vector 163 188 COORDINATES AND TENSORS Hence aa and ... axn and Adding the first two equations and subtracting the last one from the result and after some rearrangement of indices we obtain (10.197) I0 COORDINATES and TENSORS Starti...

Ngày tải lên: 13/08/2014, 09:21