1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

328 Chapter 5 Higher Dimensions and Other Coordinates Figure 9 Nodal curves: The curves in these graphs represent solutions of φ ...

5.8 Some Applications of Bessel Functions 329 Figure 10 Exercise 10. ofa)? What are the radii of the circles that are the nodal ...

330 Chapter 5 Higher Dimensions and Other Coordinates Several problems in which the Bessel functions play an important role foll ...

5.8 Some Applications of Bessel Functions 331 Thea’s andb’s are determined from Eqs. (4) and (5) by using the orthogonality rela ...

332 Chapter 5 Higher Dimensions and Other Coordinates Thus in order to satisfy Eq. (20), we must haveB=0. It is possible to show ...

5.8 Some Applications of Bessel Functions 333 ( x^3 X′ )′ +λ^2 x^3 X= 0 , a<x<b, (31) X(a)= 0 , X(b)= 0 , (32) Y′′−λ^2 Y= ...

334 Chapter 5 Higher Dimensions and Other Coordinates EXERCISES 1.Find the general solution of the differential equation ( xnφ′ ...

5.9 Spherical Coordinates; Legendre Polynomials 335 5.9 Spherical Coordinates; Legendre Polynomials After the Cartesian and cyli ...

336 Chapter 5 Higher Dimensions and Other Coordinates of variables: 1 ρ^2 { ∂ ∂ρ ( ρ^2 ∂u ∂ρ ) +^1 sin(φ) ∂ ∂φ ( sin(φ)∂u ∂φ )} ...

5.9 Spherical Coordinates; Legendre Polynomials 337 Solutions of the differential equation are usually found by the power series ...

338 Chapter 5 Higher Dimensions and Other Coordinates P 0 (x)= 1 P 1 (x)=x P 2 (x)=( 3 x^2 − 1 )/ 2 P 3 (x)=( 5 x^3 − 3 x)/ 2 P ...

5.9 Spherical Coordinates; Legendre Polynomials 339 Since the differential equation (5) is easily put into self-adjoint form, ( ...

340 Chapter 5 Higher Dimensions and Other Coordinates Theorem. If f(x)is sectionally smooth on the interval− 1 <x< 1 ,then ...

5.9 Spherical Coordinates; Legendre Polynomials 341 Next, move the last term to the left-hand member of the equation to find (n+ ...

342 Chapter 5 Higher Dimensions and Other Coordinates Example. Let f(x)= {− 1 , − 1 <x<0, 1 , 0 <x<1. The Legendre s ...

5.9 Spherical Coordinates; Legendre Polynomials 343 Figure 14 The nodal curves of the zonal harmonics are the parallels (φ=const ...

344 Chapter 5 Higher Dimensions and Other Coordinates Show that the coefficients are all zero afteranifμ^2 =n(n+ 1 ). 2.Derive t ...

5.10 Some Applications of Legendre Polynomials 345 5.10 Some Applications of Legendre Polynomials In this section we follow thro ...

346 Chapter 5 Higher Dimensions and Other Coordinates The general solution of the partial differential equation that is bounded ...

5.10 Some Applications of Legendre Polynomials 347 Now, a series of constant multiples of product solutions is the most general ...