This ODE arises in many physical systems that we shall investigate. + ( + 1)u = 0. (λ + s)x λ + s + ( + 1) a λ. (s + 1)(s + 2) a 0


 Archibald Scott
 1 years ago
 Views:
Transcription
1 Legendre equation This ODE arises in many physical systems that we shall investigate We choose We then have Substitution gives ( x 2 ) d 2 u du 2x 2 dx dx + ( + )u u x s a λ x λ a du dx λ a λ (λ + s)x λ + s λ d 2 u dx a 2 λ (λ + s)(λ + s )x λ + s 2 λ a λ (λ + s)(λ + s )x λ + s 2 λ a λ (λ + s)(λ + s )x λ + s λ 2 a λ (λ + s)x λ + s + ( + ) a λ x λ + s λ [ a s(s ) ]x s 2 + [ a s(s + ) ]x s + a 2 (s + )(s + 2) a (s(s ) + 2s ( + )) λ [ ]x s [ a n+2 (s + n + 2)(s + n + ) a n ((s + n)(s + n ) + 2(s + n) ( + )) ]x s+n +...
2 or a s(s ) s, a s(s + ) s, a and s, a a n+2 (s + n + 2)(s + n + ) a n ((s + n)(s + n ) + 2(s + n) ( + )) or (s + n)(s + n ) + 2(s + n) ( + ) a n+2 (s + n + 2)(s + n + ) For s we have For s we have a n+2 a n+2 n(n + ) ( + ) (n + 2)(n + ) a n (n + 2)(n + ) ( + ) (n + 3)(n + 2) a n a n The even and odd solutions are then ( + ) ( 2) ( + )( + 3) u x 2 + x ! 4! ( )( + 2) ( 3)( )( + 2)( + 4) u 2 x x 3 + 3! 5! x
3 As before the other case does not contribute anything new. If is an integer, then the series terminate at a finite number of terms and the solutions are polynomials. We have λ u, u 2 infinite series λ u infinite series, u 2 x λ 2 u 3x 2, u 2 infinite series λ 3 u infinite series, u 2 x 5 3 x3 λ 4 u x x4, u 2 infinite series λ 5 u infinite series, u 2 x 4 3 x x5 If we multiply a solution by a constant, then it is still a solution. 3
4 If we multiply each polynomial above by a constant so that they have the value at x, then they are called the Legendre polynomials P, where In general we write P where P P x P 2 ( 2 3x2 ) P 3 ( 2 5x3 3x) P 4 ( 8 35x4 3x 2 + 3) P 5 ( 8 63x5 7x 3 + 5x) N n ( ) n (2 2n)! 2 n!( n)!( 2n)! x 2n N / 2 even ( ) / 2 odd The infinite series and the polynomials generate Legendre's functions of the second kind Q where Q u ()u 2 u 2 ()u even odd 4
5 and the most general solution of Legendre's equations is u c P + c 2 Q All Legendre's functions of the second kind are singular at x; this will be important in physical problems. If we make the substitution x cosθ Legendre's equation becomes d du sinθ sinθ dθ dθ + ( + )u which is the form that will arise most often when we solve problems in sphericalpolar coordinates. Hermite Equation Now we turn our attention to Hermite' equation d 2 u du 2x + 2αu, α nonnegative integer 2 dx dx As we shall see, the solutions are polynomials called Hermite polynomials. Legendre and Hermite polynomials are the first two examples of a large class of special functions that play a major role in the solutions of partial differential equations representing physical systems. We choose 5
6 We then have u x s a λ x λ a du dx λ a λ (λ + s)x λ + s λ d 2 u dx a 2 λ (λ + s)(λ + s )x λ + s 2 λ Substitution gives a λ (λ + s)(λ + s )x λ + s 2 2 a λ λ λ (λ + s)x λ + s + 2α a λ x λ + s [ a s(s ) ]x s 2 + [ a s(s + ) ]x s + [ a 2 (s + )(s + 2) a (s 2α) ]x s [ a n+2 (s + n + 2)(s + n + ) a n ((s + n) 2α) ]x s+n +... or a s(s ) s, a s(s + ) s, a and s, a a n+2 (s + n + 2)(s + n + ) a n ((s + n) 2α) or a n+2 (s + n) 2α (s + n + 2)(s + n + ) a n λ 6
7 For s we have For s we have a n+2 a n+2 n 2α (n + 2)(n + ) a n (+ n) 2α (n + 3)(n + 2) a n The even and odd solutions are then u 2α 2! x α(2 α) x ! 2( α) u 2 x + x ( α)(3 α) x ! 5! As before the other case does not contribute anything new. Clearly the infinite series terminate if α is an integer. The Hermite polynomials are H H 2x H 2 4x 2 2 H 3 8x 3 2x or in general 7
8 where H n n! N k ( ) k (2x) n 2k k!(n 2k)! N n / 2 n even (n ) / 2 n odd The Hermite polynomials appear in the harmonic oscillator solution of the Schrodinger equation as shown below. The Schrodinger equation for the dimensional harmonic oscillator with a potential function V 2 kx2 is given by 2 2m d 2 ψ dx kx2 ψ Eψ The standard ODE solution of this equations goes as follows. Let ξ αx α 4 mk 2 λ 2E ω k /2 ω m This gives the new equation d 2 ψ dξ 2 + λ ξ 2 ( )ψ 8
9 ξ ± If we consider the behavior of the equation as we can neglect and the equation becomes d 2 ψ dξ ξ 2 ψ 2 which says that the large ξ behavior of the solution is ψ e ξ2 /2 This suggests that we try a solution of the form which should remove the large ψ H (ξ)e ξ2 /2 ξ behavior from the equation. We get the equation H '' 2ξH '+ (λ )H This is Hermite's equation and provided we choose λ 2n +. If we do not terminate the series the solutions are not square integrable and cannot represent wave functions. The series solutions we found earlier have the recursion relation a k +2 (2k + λ) (k + 2)(k + ) a k If the series does not terminate, then its dominant asymptotic behavior can be inferred from the coefficients of its high terms λ 9
10 lim k a k +2 a k 2 k ξ n e ξ2 This ratio is the same as that of the series for with any finite value of n. This means that the solution is terminated. ψ H (ξ)e ξ2 /2 will not be squareintegrable unless the series So we terminate the series and the solutions are the Hermite polynomials. For this choice of λ with associated wave functions we get energy eigenvalues E n ω n + 2 ψ n H n (αx)e α 2 x 2 /2 That is why we are studying all these special equations  we will then recognize them when we solve the equations for real physical systems and can just quote our earlier results. Gamma or Factorial Function Consider the integrals below: e α x dx α, xe α x dx, x 2 e α x dx 2!, x 3 e α x dx 3! α 2 α 3 α 4 H n
11 or If we let α, then x n e α x dx n! α n+ n! x n e x dx where we have defined n! n(n )(n 2)...(2)() and! Generalizing this result we have the Gamma Function Γ(p) and for n integer Γ(n) Now Γ(p + ) x p e x dx for p > x n e x dx (n )! Γ(n + ) n! x p e x dx for p > x p e x This is a recursion relation for the + p x p e x dx Γ function. p x p e x dx pγ(p)
12 What about negative numbers? We can use the recursion relation: Γ(p) p Γ(p + ) which gives Now Γ(.5).5 Γ(.5 + ) 2Γ(.5) Γ(.5).5 Γ(.5 + ) 2 3 Γ(.5) 4 3 Γ(.5) Γ(.5) t /2 e t dt t y 2 dt 2ydy Γ(.5) y e y2 2ydy 2 e y2 dy [ Γ(.5) ] 2 4 e x2 dx e y2 dy 4 e (x2 + y 2 ) dxdy 4 e r2 rdrdφ π /2 2π e r2 rdr 2π d 2 e r2 π Γ(.5) π In addition, since Γ() we have 2
13 lim p Γ(p + ) p Γ( p) for all negative integers Laguerre Equation  The Laguerre equation takes the form We choose We then have Substitution gives x d 2 u du + ( x) 2 dx dx + Nu u x s a λ x λ a du dx λ a λ (λ + s)x λ + s λ d 2 u dx a 2 λ (λ + s)(λ + s )x λ + s 2 λ a λ (λ + s)(λ + s )x λ + s λ a λ (λ + s)x λ + s λ + a λ (λ + s)x λ + s + N a λ x λ + s [ a s(s ) + a s]x s + [ a s(s + ) a s + a (s + ) + Na ]x s +... λ + [ a n+ (s + n + )(s + n) a n (s + n) + a n+ (s + n + ) + Na n ]x s+n +... λ 3
14 or a s(s ) + a s s 2 s a n+ (s + n + )(s + n) a n (s + n) + a n+ (s + n + ) + Na n or a n+ For s we have or a ( ) (s + n) N (s + n + )(s + n) + (s + n + ) a n a n+ n N (n + ) 2 a n (s + n) N (s + n + ) 2 a n N, a (!) 2 2 N N(N ) a 2 2 ( ) 2, a (2!) 2 a k ( ) k N(N )...(N k + ) (k!) 2 ( ) k k! N! k!(n k)! ( )k N k k! It is clear that if k > N, then u L N + a k. Thus we get N N k N ( ) k x k ( ) k k! k k N k k! x k polynomial 4
15 These are the Laguerre polynomials: L L x L 2 2x + x2 They will appear in the hydrogen atom solution of the Schrodinger equation. Before proceeding to the final example, namely, Bessel s equation, let us digress to study generating functions. Generating Functions 2! Legendre Polynomials Redux  Let us consider the function φ(x,h) ( 2xh + h 2 ), h < /2 h f (power series) Differentiation gives ( x 2 ) 2 φ φ 2x 2 x x + h 2 (hφ) h 2 If we substitute a power series we get 5
16 ( x 2 ) h f '' 2xh f ' + h ( + )h f ( x 2 ) f '' 2xf ' + ( + ) f This says that the coefficient of each power of h must separately equal zero. We get ( x 2 ) f '' 2xf ' + ( + ) f This is Legendre s equation. So the solutions are f P Legendre polynomials and φ(x,h) ( 2xh + h 2 ) P h /2 Generating function for the Legendre polynomials Let us generate the first couple of polynomials. Let φ(x,h) ( y) /2 + 2 y y xh x2 2 h P + hp + h 2 P y 2xh h 2. Then 6
17 or P, P x, P x2 2 We can also write Recursion Relations P! x φ(x,h) h Legendre polynomials of different P xp via recursion or recurrence relations. Now consider We have φ(x,t) φ(x,t) t Rearranging we get (degree) are related to each other, i.e. ( 2xt + t 2 ) t n P /2 n n x t ( 2xt + t 2 ) nt n P 3/2 n n n ( x t)φ(x,t) ( 2xt + t 2 ) nt n P n x t n ( ) t n P n (2n + )tt n P n P + t n (np n + (n + )P n+ ) n Equating coefficients of t n n we have the recursion relation 7
18 Therefore, given that (2n + )xp n np n + (n + )P n+ P and P x 3P P + 2P 2 P 2 3x 2 In a similar manner we can find Orthogonality Relations We have np n xp' n P' n φ 2 (x,t)dx ( + t 2 2tx) dx where we have used n m t n+ m Equating coefficients we get t P n P m dx n n( + h) ( ) n+ we get ( n( + t) n( t) ) 2 2k + t 2k h n n k P n P m dx 2 2n + δ nm 8
19 Thus, it is clear that generating functions are very powerful tools. Hermite Polynomials As we have seen, families of functions can be defined by ODEs. We have also seen that they can be defined by means of a generating function. Finally, they can be defined by a differential formula of the form such as the Rodriguez formula for Hermite polynomials. H n ( ) n e x2 which gives H d n 2 dx n e x H 2x H 2 4x 2 2, n,,2,3,... We can get the generating function from this formula by introducing a dummy variable t as follows. H n e x2 Earlier we saw that Therefore, we have n 2 t n e (x t ) t P! x φ(x,h) n 2 t n e2 x t h t 9
20 n t n e 2 x t2 H n n! φ(x,t) generating function In the same way as before we can derive recursion and orthogonality relations. H n+ 2xH n 2nH n H ' n 2nH n H n+ 2xH n + H ' n Differentiating this last relation we get H ' n+ 2H n 2xH ' n + H '' n H '' n 2xH ' n 2nH n which is the Hermite ODE. The solutions are the Hermite polynomials. We also have the orthogonality relations Laguerre Polynomials A similar procedure gives e x2 H n H m dx 2 n n! πδ nm L n n! ex d n dx n φ(x,t) t e xt t ( e x x n ) n L n t n 2
21 (n + )L n+ (2n + )L n xl n nl n xl ' n nl n nl n Differentiation then gives the Laguerre ODE xl '' n + ( x)l ' n + nl n and finally, we have the orthogonality relation Bessel's Equation e x L n L m dx δ nm We now consider the one of the most important equation in physics. ( ) x 2 d 2 dx 2 + x d dx + x2 µ 2 We choose u x s a λ x λ a We then have du dx u L L( x) exists even and odd solutions λ a λ (λ + s)x λ + s λ d 2 u dx a 2 λ (λ + s)(λ + s )x λ + s 2 λ 2
22 Substitution gives or or a λ (λ + s)(λ + s )x λ + s + λ a s(s ) + a s µ 2 a x s + a λ (λ + s)x λ + s λ + a λ x λ + s+2 µ 2 a λ x λ + s λ a s(s + ) + a (s + ) µ 2 a x s a n (s + n)(s + n ) + a n (s + n) + a n 2 µ 2 a n x s+n +... a s(s ) + a s µ 2 a s 2 µ 2 s ±µ a s(s + ) + a (s + ) µ 2 a (s 2 + 2s + µ 2 )a a a n (s + n)(s + n ) + a n (s + n) + a n 2 µ 2 a nn or a n (s + n )(s + n) + (s + n) µ 2 a n 2 λ (s + n) 2 µ 2 a n 2 a Since this last relation implies that all odd coefficients. 22
23 For s µ, we have a n (n + µ) 2 µ 2 a n 2 or after some algebra a 2n and the solution is J µ ( ) n (2) 2n+ µ n!γ(µ + n + ), a ( ) n x 2n+ µ (2) 2n+ µ n!γ(n + µ + ) n 2 µ Γ(µ + ) µ which is the Bessel function of the first kind of order. For s µ, we have a n (n µ) 2 µ 2 a n 2 or after some algebra and the solution is J µ ( ) n a 2n (2) 2n µ n!γ(n µ + ) ( ) n x 2n µ (2) 2n µ n!γ(n µ + ) n 23
24 which is the Bessel function of the second kind of order which will be important in many physical problems. The most general solution is then u aj µ + bj µ µ. It is singular at x, Bessel functions appear as solutions in an amazing number of physical systems. Bessel Function Properties Orthogonal polynomials such as Legendre, Hermite and Laguerre polynomials are wellsuited(can be used as basis functions) for expansions of functions or the description of physical system that are localized near the origin of coordinates. Like Taylor expansions, the further we go away from the origin, the more terms we need. Bessel functions help us get around this difficulty. The Bessel function J n of integral order n satisfies the ODE x 2 J '' n + xj ' n + (x 2 n 2 )J n The generating function for Bessel functions is G(x,t) exp 2 x t t n t n J n 24
25 To explicitly obtain the J n we expand G(x,t) in powers of t. exp 2 x t t exp 2 xt exp x 2 r,s ( ) s x 2 r + s t r s r!s! t r x r t r 2 r! x 2 s r t s r! We change the summation indices to n rs or r n+s and r+s n+2s which gives or n+2s x G(x,t) ( ) s 2 J n n s n+2s t n ( ) s x (n + s)!s! 2 for n s J n x n 2 n! x2 4 For n < a similar analysis gives J n (n + s)!s! t n J n n! (n + )! + x2 4 2s n 2! (n + 2)!... 2 ( ) s x (s n )!s! 2 for n < s n since Γ(n) for the negative integers. 25
26 We also have J n ( ) n J n for integer n. The generating function gives the recursion relation ( 2 J n + J n+ ) n x J n ( ) J ' n 2 J J n n+ Adding and subtracting we get the ladder operators Since R n J n L n J n n x d dx n x + d dx J n J n+ J n J n L n+ R n J n L n+ J n+ J n we get (L n+ R n )J n which is Bessel s equation. 26
27 One can easily verify that the expansion for J n can have substituted for n where v is any real number and then direct differentiation shows that J ν satisfies Bessel s equation with v in place of n. The factorials in the expansion do not present a problem since they are defined for noninteger values by z! Γ(z + ) e t t z dt In many cases, instead of the J ν functions, one uses the Neuman functions defined by N ν cosνπ J J ν ν sinνπ N ν is linearly independent of J ν. This means it is also a solution of Bessel s equation. It is the second linearly independent solution, i.e., This also holds for W Wronskian J ν N ' v J ' ν N ν ( J ν J ' ν + J ' ν J ν ) / sinνπ ν n 2 independent of ν π x which says the two solutions are linearly independent. v We also have the relations d dx x p J p ( ) x p J p x J p(ax)j p (bx)dx a b 2 J 2 p+(a) 2 J 2 p (a) 2 J ' 2 p(a) a b 27
28 Relationship of the Cylindrical and Spherical Bessel Functions The spherical Bessel functions are defined by: j (z) η (z) π 2z π 2z /2 /2 J +/2 (z) N +/2 (z) Some simple ones are: In general: j (z) sin z z η (z) cos z j (z) j ' (z) sin z z 2 j 2 (z) j z j ' (z) j + j z j cos z z 3 z 3 z sin z 3 z cos z 2 Very important in quantum mechanics. j ( + ) j + (2 + ) j ' 28
29 Some Odd Topics Beta Function [] B(p,q) B(q, p) [2] x y / a B(p,q) y p (a y) q dy [3] x sin 2 θ B(p,q) 2 (sinθ) p (cosθ) q dθ a π /2 [4] x y + y B(p,q) y p [5] B(p,q) Γ(p)Γ(q) Γ(p + q) dy q (+ y) Error Function x erf 2 π e t 2 dt Stirling's Formula (Factorial function for large arguments) From the definition Γ(p + ) p! x p e x dx e p nx x dx Now let x p + y p : x y p, dx pdy 29
30 We then have Now so that or more exactly p p! e p n( p+ y p ) p y p n(p + y p) np + y p pdy np + n + y For p very large n(p + y p) np + n + y p np + p y p y2 2 p +... p p! e p np p p e y2 /2 dy p p e p p e y2 /2 dy e y2 /2 dy p p p e p p e y2 /2 dy p p e p 2π p for large p Γ(p + ) p! p p e p 2π p + 2 p p And now for something completely different... Expansions in Orthonormal Functions GramSchmidt in place of ODEs (partial repeat of earlier material from week ) 3
31 Definitions:. Expansion interval [a,b] : the variable x is confined to the interval a x 2. Given two functions f and g defined on the interval [a,b], the inner product of f and g is denoted by (f,g) and is given by b a ( f, g) f * gdx Over the space of complex functions. Note that order is important since ( f, g) ( g, f ) *. 3. If (f,g) the functions f and g are orthogonal. 4. The norm of a function is (f,f). If the norm is finite, the function is squareintegrable. If the function has norm, then it is normalized. Any squareintegrable function can be normalized by multiplication by a constant. ϕ,ϕ 2, A set of functions defined on the interval [a,b] is said to be orthonormal if ( ϕ i,ϕ ) j δ ij If we have an orthonormal set of functions on the interval [a,b] then we can write any other squareintegrable function f in terms of that set by f a j ϕ j where ( ϕ k, f ) a j (ϕ k,ϕ j ) a j δ jk j If this is true for any function f, then set of functions and is called a basis set. j j a k ϕ,ϕ 2,... is complete 3
32 GramSchmidt Orthogonalization Method. It turns out that to be a basis set, a set of functions only has to be linearly independent. Linear independence implies that there exists no relation n c λ f λ λ c λ valid for all x, except the trivial one of all. They do not have to be orthonormal...that is only a convenience. Suppose we have a set of n linearly independent, square integrable functions f λ λ,2, 3, 4,...,n ϕ,ϕ 2,...,ϕ n on the interval [a,b]. How do we construct an orthonormal set? There is a systematic process for doing this. It goes as follows:. Choose ϕ N f where N ( f, f ) 2. Choose ϕ 2 as a linear combination of f and f 2 that is orthogonal to ϕ and normalized to. ϕ 2 [ f 2 (ϕ, f 2 )ϕ ] where N 2 ( f 2, f 2 ) (ϕ, f 2 ) 2 N 2 We have just subtracted off the nonorthogonal part! 32
33 3. This generalizes as k ϕ k [ f k (ϕ j, f k )ϕ j ] where N k ( f k, f k ) (ϕ j, f k ) 2 N k Legendre Polynomials Redux Redux j k j Consider the set of linearly independent functions which are the nonnegative integral powers of x on the interval [,] f λ x λ λ,,2, 3, 4,...,n Using the orthogonalization procedure above we get the new set of functions ϕ f where N ( f, f ) dx 2 ϕ N N 2 ϕ [ f (ϕ, f )ϕ ] x xdx N N 2 2 x N where N ( f, f ) (ϕ, f ) 2 and in the same manner ϕ ϕ ϕ x 2 dx () 2 2 / 3 ϕ 2 x x3 3 2 x 35 8 x4 5 4 x x 33
34 Apart from normalization these are the Legendre polynomials, i.e., ϕ j 2 j + P j 2 Thus, we can construct the orthogonal polynomials using linear algebra instead of ODEs. Finally, let us look at some weird numbers that have a habit of popping up in real world systems. Bernoulli Numbers Consider the series x e x n B n n! xn We can use the relation B n d n dx n x e x x B n to determine the Bernoulli numbers. This is a difficult direct process, however. An easier method is to use the uniqueness property of the power series expansions. We can write e x x x + 2! x2 + 3! x x + 2! x + 3! x2 + 4! x
35 We then get + 2! x + 3! x2 + 4! x B + B x + 2! B2x which gives(by uniqueness or equating coefficients of like powers of x) and so on... B In general we have, 2! B + B B 2 3! B + 2! B + 2! B 2 B 2 6 n n k B 2k k We obtain B 2n+, n,2,3,... B 2n ( )n 2(2n)! p 2n, n,2,3,... (2π ) 2n p These numbers are very divergent, i.e., B , B The Bernoulli numbers appear all over the place, i.e., tan x x + 6 x x ( )n 2 2n (2 2n ) B 2n x 2n +... (2n)! 35
36 Bernoulli Functions The functions are defined by It is clear that xe xs e x n B n (s) xn n! B n Bernoulli number B n () Other useful properties are: B' n (s) nb n (s), n,2,3,... B n () ( ) n B n (), n,,2,3,... 36
PHYS 404 Lecture 1: Legendre Functions
PHYS 404 Lecture 1: Legendre Functions Dr. Vasileios Lempesis PHYS 404  LECTURE 1 DR. V. LEMPESIS 1 Legendre Functions physical justification Legendre functions or Legendre polynomials are the solutions
More informationMath221: HW# 7 solutions
Math22: HW# 7 solutions Andy Royston November 7, 25.3.3 let x = e u. Then ln x = u, x2 = e 2u, and dx = e 2u du. Furthermore, when x =, u, and when x =, u =. Hence x 2 ln x) 3 dx = e 2u u 3 e u du) = e
More informationd 1 µ 2 Θ = 0. (4.1) consider first the case of m = 0 where there is no azimuthal dependence on the angle φ.
4 Legendre Functions In order to investigate the solutions of Legendre s differential equation d ( µ ) dθ ] ] + l(l + ) m dµ dµ µ Θ = 0. (4.) consider first the case of m = 0 where there is no azimuthal
More informationS. Ghorai 1. Lecture XV Bessel s equation, Bessel s function. e t t p 1 dt, p > 0. (1)
S Ghorai 1 1 Gamma function Gamma function is defined by Lecture XV Bessel s equation, Bessel s function Γp) = e t t p 1 dt, p > 1) The integral in 1) is convergent that can be proved easily Some special
More informationTwo special equations: Bessel s and Legendre s equations. p FourierBessel and FourierLegendre series. p
LECTURE 1 Table of Contents Two special equations: Bessel s and Legendre s equations. p. 259268. FourierBessel and FourierLegendre series. p. 453460. Boundary value problems in other coordinate system.
More informationENGI 9420 Lecture Notes 1  ODEs Page 1.01
ENGI 940 Lecture Notes  ODEs Page.0. Ordinary Differential Equations An equation involving a function of one independent variable and the derivative(s) of that function is an ordinary differential equation
More informationMathematics for Chemistry: Exam Set 1
Mathematics for Chemistry: Exam Set 1 June 18, 017 1 mark Questions 1. The minimum value of the rank of any 5 3 matrix is 0 1 3. The trace of an identity n n matrix is equal to 11 0 n 3. A square matrix
More information4 Power Series Solutions: Frobenius Method
4 Power Series Solutions: Frobenius Method Now the ODE adventure takes us to series solutions for ODEs, a technique A & W, that is often viable, valuable and informative. These can be readily applied Sec.
More informationMathematics for Chemistry: Exam Set 1
Mathematics for Chemistry: Exam Set 1 March 19, 017 1 mark Questions 1. The maximum value of the rank of any 5 3 matrix is (a (b3 4 5. The determinant of an identity n n matrix is equal to (a 1 (b 1 0
More informationswapneel/207
Partial differential equations Swapneel Mahajan www.math.iitb.ac.in/ swapneel/207 1 1 Power series For a real number x 0 and a sequence (a n ) of real numbers, consider the expression a n (x x 0 ) n =
More informationMATH3383. Quantum Mechanics. Appendix D: Hermite Equation; Orthogonal Polynomials
MATH3383. Quantum Mechanics. Appendix D: Hermite Equation; Orthogonal Polynomials. Hermite Equation In the study of the eigenvalue problem of the Hamiltonian for the quantum harmonic oscillator we have
More informationChapter 4. Series Solutions. 4.1 Introduction to Power Series
Series Solutions Chapter 4 In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generation adds a new story to the old
More informationThe Sommerfeld Polynomial Method: Harmonic Oscillator Example
Chemistry 460 Fall 2017 Dr. Jean M. Standard October 2, 2017 The Sommerfeld Polynomial Method: Harmonic Oscillator Example Scaling the Harmonic Oscillator Equation Recall the basic definitions of the harmonic
More informationRelevant sections from AMATH 351 Course Notes (Wainwright): Relevant sections from AMATH 351 Course Notes (Poulin and Ingalls):
Lecture 5 Series solutions to DEs Relevant sections from AMATH 35 Course Notes (Wainwright):.4. Relevant sections from AMATH 35 Course Notes (Poulin and Ingalls): 2.2.3 As mentioned earlier in this course,
More informationSolutions: Problem Set 3 Math 201B, Winter 2007
Solutions: Problem Set 3 Math 201B, Winter 2007 Problem 1. Prove that an infinitedimensional Hilbert space is a separable metric space if and only if it has a countable orthonormal basis. Solution. If
More informationBessel s and legendre s equations
Chapter 12 Bessel s and legendre s equations 12.1 Introduction Many linear differential equations having variable coefficients cannot be solved by usual methods and we need to employ series solution method
More informationODE Homework Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation. n(n 1)a n x n 2 = n=0
ODE Homework 6 5.2. Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation y + k 2 x 2 y = 0, k a constant about the the point x 0 = 0. Find the recurrence relation;
More informationFinitedimensional spaces. C n is the space of ntuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a prehilbert space, or a unitary space) if there is a mapping (, )
More informationFinal Exam May 4, 2016
1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.
More informationSeries Solution of Linear Ordinary Differential Equations
Series Solution of Linear Ordinary Differential Equations Department of Mathematics IIT Guwahati Aim: To study methods for determining series expansions for solutions to linear ODE with variable coefficients.
More information5.4 Bessel s Equation. Bessel Functions
SEC 54 Bessel s Equation Bessel Functions J (x) 87 # with y dy>dt, etc, constant A, B, C, D, K, and t 5 HYPERGEOMETRIC ODE At B (t t )(t t ), t t, can be reduced to the hypergeometric equation with independent
More informationDifferential Equations
Differential Equations Problem Sheet 1 3 rd November 2011 FirstOrder Ordinary Differential Equations 1. Find the general solutions of the following separable differential equations. Which equations are
More informationLecture 4.6: Some special orthogonal functions
Lecture 4.6: Some special orthogonal functions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics
More informationLEGENDRE POLYNOMIALS AND APPLICATIONS. We construct Legendre polynomials and apply them to solve Dirichlet problems in spherical coordinates.
LEGENDRE POLYNOMIALS AND APPLICATIONS We construct Legendre polynomials and apply them to solve Dirichlet problems in spherical coordinates.. Legendre equation: series solutions The Legendre equation is
More informationAdvanced Computational Fluid Dynamics AA215A Lecture 2 Approximation Theory. Antony Jameson
Advanced Computational Fluid Dynamics AA5A Lecture Approximation Theory Antony Jameson Winter Quarter, 6, Stanford, CA Last revised on January 7, 6 Contents Approximation Theory. Least Squares Approximation
More informationLaplace s Equation in Cylindrical Coordinates and Bessel s Equation (I)
Laplace s Equation in Cylindrical Coordinates and Bessel s Equation I) 1 Solution by separation of variables Laplace s equation is a key equation in Mathematical Physics. Several phenomena involving scalar
More informationInner Product Spaces An inner product on a complex linear space X is a function x y from X X C such that. (1) (2) (3) x x > 0 for x 0.
Inner Product Spaces An inner product on a complex linear space X is a function x y from X X C such that (1) () () (4) x 1 + x y = x 1 y + x y y x = x y x αy = α x y x x > 0 for x 0 Consequently, (5) (6)
More informationMathematics 324 Riemann Zeta Function August 5, 2005
Mathematics 324 Riemann Zeta Function August 5, 25 In this note we give an introduction to the Riemann zeta function, which connects the ideas of real analysis with the arithmetic of the integers. Define
More informationPHYS 771, Quantum Mechanics, Final Exam, Fall 2011 Instructor: Dr. A. G. Petukhov. Solutions
PHYS 771, Quantum Mechanics, Final Exam, Fall 11 Instructor: Dr. A. G. Petukhov Solutions 1. Apply WKB approximation to a particle moving in a potential 1 V x) = mω x x > otherwise Find eigenfunctions,
More informationInfinite Series. 1 Introduction. 2 General discussion on convergence
Infinite Series 1 Introduction I will only cover a few topics in this lecture, choosing to discuss those which I have used over the years. The text covers substantially more material and is available for
More informationSeries Solutions of ODEs. Special Functions
C05.tex 6/4/0 3: 5 Page 65 Chap. 5 Series Solutions of ODEs. Special Functions We continue our studies of ODEs with Legendre s, Bessel s, and the hypergeometric equations. These ODEs have variable coefficients
More informationHilbert Space Problems
Hilbert Space Problems Prescribed books for problems. ) Hilbert Spaces, Wavelets, Generalized Functions and Modern Quantum Mechanics by WilliHans Steeb Kluwer Academic Publishers, 998 ISBN 79235239
More informationCollection of formulae Quantum mechanics. Basic Formulas Division of Material Science Hans Weber. Operators
Basic Formulas 1711 Division of Material Science Hans Weer The de Broglie wave length λ = h p The Schrödinger equation Hψr,t = i h t ψr,t Stationary states Hψr,t = Eψr,t Collection of formulae Quantum
More informationNotes on Special Functions
Spring 25 1 Notes on Special Functions Francis J. Narcowich Department of Mathematics Texas A&M University College Station, TX 778433368 Introduction These notes are for our classes on special functions.
More informationwhich arises when we compute the orthogonal projection of a vector y in a subspace with an orthogonal basis. Hence assume that P y = A ij = x j, x i
MODULE 6 Topics: GramSchmidt orthogonalization process We begin by observing that if the vectors {x j } N are mutually orthogonal in an inner product space V then they are necessarily linearly independent.
More informationQuantum Harmonic Oscillator
Quantum Harmonic Oscillator Chapter 13 P. J. Grandinetti Chem. 4300 Oct 20, 2017 P. J. Grandinetti (Chem. 4300) Quantum Harmonic Oscillator Oct 20, 2017 1 / 26 Kinetic and Potential Energy Operators Harmonic
More informationSpherical Harmonics on S 2
7 August 00 1 Spherical Harmonics on 1 The LaplaceBeltrami Operator In what follows, we describe points on using the parametrization x = cos ϕ sin θ, y = sin ϕ sin θ, z = cos θ, where θ is the colatitude
More informationBessel Functions Michael Taylor. Lecture Notes for Math 524
Bessel Functions Michael Taylor Lecture Notes for Math 54 Contents 1. Introduction. Conversion to first order systems 3. The Bessel functions J ν 4. The Bessel functions Y ν 5. Relations between J ν and
More informationHilbert Spaces. Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space.
Hilbert Spaces Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space. Vector Space. Vector space, ν, over the field of complex numbers,
More informationOn an Eigenvalue Problem Involving Legendre Functions by Nicholas J. Rose North Carolina State University
On an Eigenvalue Problem Involving Legendre Functions by Nicholas J. Rose North Carolina State University njrose@math.ncsu.edu 1. INTRODUCTION. The classical eigenvalue problem for the Legendre Polynomials
More informationPhysics 342 Lecture 23. Radial Separation. Lecture 23. Physics 342 Quantum Mechanics I
Physics 342 Lecture 23 Radial Separation Lecture 23 Physics 342 Quantum Mechanics I Friday, March 26th, 2010 We begin our spherical solutions with the simplest possible case zero potential. Aside from
More informationPower Series Solutions to the Bessel Equation
Power Series Solutions to the Bessel Equation Department of Mathematics IIT Guwahati The Bessel equation The equation x 2 y + xy + (x 2 α 2 )y = 0, (1) where α is a nonnegative constant, i.e α 0, is called
More informationVector Spaces. Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms.
Vector Spaces Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms. For each two vectors a, b ν there exists a summation procedure: a +
More informationElectromagnetism HW 1 math review
Electromagnetism HW math review Problems 5 due Mon 7th Sep, 6 due Mon 4th Sep Exercise. The LeviCivita symbol, ɛ ijk, also known as the completely antisymmetric rank3 tensor, has the following properties:
More informationBackground and Definitions...2. Legendre s Equation, Functions and Polynomials...4 Legendre s Associated Equation and Functions...
Legendre Polynomials and Functions Reading Problems Outline Background and Definitions...2 Definitions...3 Theory...4 Legendre s Equation, Functions and Polynomials...4 Legendre s Associated Equation and
More informationScientific Computing
2301678 Scientific Computing Chapter 2 Interpolation and Approximation Paisan Nakmahachalasint Paisan.N@chula.ac.th Chapter 2 Interpolation and Approximation p. 1/66 Contents 1. Polynomial interpolation
More informationReview of Power Series
Review of Power Series MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Introduction In addition to the techniques we have studied so far, we may use power
More informationPart 3.3 Differentiation Taylor Polynomials
Part 3.3 Differentiation 3..3.1 Taylor Polynomials Definition 3.3.1 Taylor 1715 and Maclaurin 1742) If a is a fixed number, and f is a function whose first n derivatives exist at a then the Taylor polynomial
More informationSome Fun with Divergent Series
Some Fun with Divergent Series 1. Preliminary Results We begin by examining the (divergent) infinite series S 1 = 1 + 2 + 3 + 4 + 5 + 6 + = k=1 k S 2 = 1 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 + = k=1 k 2 (i)
More informationMath 1B Final Exam, Solution. Prof. Mina Aganagic Lecture 2, Spring (6 points) Use substitution and integration by parts to find:
Math B Final Eam, Solution Prof. Mina Aganagic Lecture 2, Spring 20 The eam is closed book, apart from a sheet of notes 8. Calculators are not allowed. It is your responsibility to write your answers clearly..
More informationPhysics 116C Solutions to the Practice Final Exam Fall The method of Frobenius fails because it works only for Fuchsian equations, of the type
Physics 6C Solutions to the Practice Final Exam Fall 2 Consider the differential equation x y +x(x+)y y = () (a) Explain why the method of Frobenius method fails for this problem. The method of Frobenius
More informationLECTURE 10: REVIEW OF POWER SERIES. 1. Motivation
LECTURE 10: REVIEW OF POWER SERIES By definition, a power series centered at x 0 is a series of the form where a 0, a 1,... and x 0 are constants. For convenience, we shall mostly be concerned with the
More informationAsymptotics of Integrals of. Hermite Polynomials
Applied Mathematical Sciences, Vol. 4, 010, no. 61, 04056 Asymptotics of Integrals of Hermite Polynomials R. B. Paris Division of Complex Systems University of Abertay Dundee Dundee DD1 1HG, UK R.Paris@abertay.ac.uk
More informationQuantum MechanicsI Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras. Lecture  21 SquareIntegrable Functions
Quantum MechanicsI Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture  21 SquareIntegrable Functions (Refer Slide Time: 00:06) (Refer Slide Time: 00:14) We
More informationIntroduction to SturmLiouville Theory and the Theory of Generalized Fourier Series
CHAPTER 5 Introduction to SturmLiouville Theory and the Theory of Generalized Fourier Series We start with some introductory examples. 5.. Cauchy s equation The homogeneous EulerCauchy equation (Leonhard
More informationProblem Set 8 Mar 5, 2004 Due Mar 10, 2004 ACM 95b/100b 3pm at Firestone 303 E. Sterl Phinney (2 pts) Include grading section number
Problem Set 8 Mar 5, 24 Due Mar 1, 24 ACM 95b/1b 3pm at Firestone 33 E. Sterl Phinney (2 pts) Include grading section number Useful Readings: For Green s functions, see class notes and refs on PS7 (esp
More informationLegendre s Equation. PHYS Southern Illinois University. October 18, 2016
Legendre s Equation PHYS 500  Southern Illinois University October 18, 2016 PHYS 500  Southern Illinois University Legendre s Equation October 18, 2016 1 / 11 Legendre s Equation Recall We are trying
More informationPower series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) a n z n. n=0
Lecture 22 Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) Recall a few facts about power series: a n z n This series in z is centered at z 0. Here z can
More informationQualification Exam: Mathematical Methods
Qualification Exam: Mathematical Methods Name:, QEID#41534189: August, 218 Qualification Exam QEID#41534189 2 1 Mathematical Methods I Problem 1. ID:MM12 Solve the differential equation dy + y = sin
More informationPage 404. Lecture 22: Simple Harmonic Oscillator: Energy Basis Date Given: 2008/11/19 Date Revised: 2008/11/19
Page 404 Lecture : Simple Harmonic Oscillator: Energy Basis Date Given: 008/11/19 Date Revised: 008/11/19 Coordinate Basis Section 6. The OneDimensional Simple Harmonic Oscillator: Coordinate Basis Page
More informationConnection Formula for Heine s Hypergeometric Function with q = 1
Connection Formula for Heine s Hypergeometric Function with q = 1 Ryu SASAKI Department of Physics, Shinshu University based on arxiv:1411.307[mathph], J. Phys. A in 48 (015) 11504, with S. Odake nd Numazu
More informationMath 334 A1 Homework 3 (Due Nov. 5 5pm)
Math 334 A1 Homework 3 Due Nov. 5 5pm No Advanced or Challenge problems will appear in homeworks. Basic Problems Problem 1. 4.1 11 Verify that the given functions are solutions of the differential equation,
More informationChapter 1. Preliminaries. The purpose of this chapter is to provide some basic background information. Linear Space. Hilbert Space.
Chapter 1 Preliminaries The purpose of this chapter is to provide some basic background information. Linear Space Hilbert Space Basic Principles 1 2 Preliminaries Linear Space The notion of linear space
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at krm@maths.uq.edu.au Contents 1 LINEAR EQUATIONS
More informationNonlinear Integral Equation Formulation of Orthogonal Polynomials
Nonlinear Integral Equation Formulation of Orthogonal Polynomials Eli BenNaim Theory Division, Los Alamos National Laboratory with: Carl Bender (Washington University, St. Louis) C.M. Bender and E. BenNaim,
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Corrected Version, 7th April 013 Comments to the author at keithmatt@gmail.com Chapter 1 LINEAR EQUATIONS 1.1
More informationCYK\2010\PH402\Mathematical Physics\Tutorial Find two linearly independent power series solutions of the equation.
CYK\010\PH40\Mathematical Physics\Tutorial 1. Find two linearly independent power series solutions of the equation For which values of x do the series converge?. Find a series solution for y xy + y = 0.
More informationTaylor Series and Asymptotic Expansions
Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.
More informationLecture 4: Series expansions with Special functions
Lecture 4: Series expansions with Special functions 1. Key points 1. Legedre polynomials (Legendre functions) 2. Hermite polynomials(hermite functions) 3. Laguerre polynomials (Laguerre functions) Other
More information(a) Determine the general solution for φ(ρ) near ρ = 0 for arbitary values E. (b) Show that the regular solution at ρ = 0 has the series expansion
Problem 1. Curious Wave Functions The eigenfunctions of a D7 brane in a curved geometry lead to the following eigenvalue equation of the Sturm Liouville type ρ ρ 3 ρ φ n (ρ) = E n w(ρ)φ n (ρ) w(ρ) = where
More informationProblem 1A. Find the volume of the solid given by x 2 + z 2 1, y 2 + z 2 1. (Hint: 1. Solution: The volume is 1. Problem 2A.
Problem 1A Find the volume of the solid given by x 2 + z 2 1, y 2 + z 2 1 (Hint: 1 1 (something)dz) Solution: The volume is 1 1 4xydz where x = y = 1 z 2 This integral has value 16/3 Problem 2A Let f(x)
More informationLinearization coefficients for orthogonal polynomials. Michael Anshelevich
Linearization coefficients for orthogonal polynomials Michael Anshelevich February 26, 2003 P n = monic polynomials of degree n = 0, 1,.... {P n } = basis for the polynomials in 1 variable. Linearization
More informationDegree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m. 11:30 a.m.
Degree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m. 11:30 a.m. Candidates should submit answers to a maximum of four
More informationAPPENDIX B GRAMSCHMIDT PROCEDURE OF ORTHOGONALIZATION. Let V be a finite dimensional inner product space spanned by basis vector functions
301 APPENDIX B GRAMSCHMIDT PROCEDURE OF ORTHOGONALIZATION Let V be a finite dimensional inner product space spanned by basis vector functions {w 1, w 2,, w n }. According to the GramSchmidt Process an
More informationClassnotes  MA Series and Matrices
Classnotes  MA2 Series and Matrices Department of Mathematics Indian Institute of Technology Madras This classnote is only meant for academic use. It is not to be used for commercial purposes. For suggestions
More informationSecondorder ordinary differential equations Special functions, SturmLiouville theory and transforms R.S. Johnson
Secondorder ordinary differential equations Special functions, SturmLiouville theory and transforms R.S. Johnson R.S. Johnson Secondorder ordinary differential equations Special functions, SturmLiouville
More information1 Ordinary points and singular points
Math 70 honors, Fall, 008 Notes 8 More on series solutions, and an introduction to \orthogonal polynomials" Homework at end Revised, /4. Some changes and additions starting on page 7. Ordinary points and
More informationSample Quantum Chemistry Exam 2 Solutions
Chemistry 46 Fall 7 Dr. Jean M. Standard Name SAMPE EXAM Sample Quantum Chemistry Exam Solutions.) ( points) Answer the following questions by selecting the correct answer from the choices provided. a.)
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,
More informationNotes for Expansions/Series and Differential Equations
Notes for Expansions/Series and Differential Equations In the last discussion, we considered perturbation methods for constructing solutions/roots of algebraic equations. Three types of problems were illustrated
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationHigherorder ordinary differential equations
Higherorder ordinary differential equations 1 A linear ODE of general order n has the form a n (x) dn y dx n +a n 1(x) dn 1 y dx n 1 + +a 1(x) dy dx +a 0(x)y = f(x). If f(x) = 0 then the equation is called
More informationwhich implies that we can take solutions which are simultaneous eigen functions of
Module 1 : Quantum Mechanics Chapter 6 : Quantum mechanics in 3D Quantum mechanics in 3D For most physical systems, the dynamics is in 3D. The solutions to the general 3d problem are quite complicated,
More information1 Solutions in cylindrical coordinates: Bessel functions
1 Solutions in cylindrical coordinates: Bessel functions 1.1 Bessel functions Bessel functions arise as solutions of potential problems in cylindrical coordinates. Laplace s equation in cylindrical coordinates
More informationMAT Linear Algebra Collection of sample exams
MAT 342  Linear Algebra Collection of sample exams Ax. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system
More informationVectors in Function Spaces
Jim Lambers MAT 66 Spring Semester 1516 Lecture 18 Notes These notes correspond to Section 6.3 in the text. Vectors in Function Spaces We begin with some necessary terminology. A vector space V, also
More informationSpecial Functions. SMS 2308: Mathematical Methods
Special s Special s SMS 2308: Mathematical Methods Department of Computational and Theoretical Sciences, Kulliyyah of Science, International Islamic University Malaysia. Sem 1 2014/2015 Factorial Bessel
More informationIntroduction to orthogonal polynomials. Michael Anshelevich
Introduction to orthogonal polynomials Michael Anshelevich November 6, 2003 µ = probability measure on R with finite moments m n (µ) = R xn dµ(x)
More informationORTHOGONAL POLYNOMIALS
ORTHOGONAL POLYNOMIALS 1. PRELUDE: THE VAN DER MONDE DETERMINANT The link between random matrix theory and the classical theory of orthogonal polynomials is van der Monde s determinant: 1 1 1 (1) n :=
More informationMath Final Exam Review
Math  Final Exam Review. Find dx x + 6x +. Name: Solution: We complete the square to see if this function has a nice form. Note we have: x + 6x + (x + + dx x + 6x + dx (x + + Note that this looks a lot
More informationRecall that any inner product space V has an associated norm defined by
Hilbert Spaces Recall that any inner product space V has an associated norm defined by v = v v. Thus an inner product space can be viewed as a special kind of normed vector space. In particular every inner
More informationIntroduction to Signal Spaces
Introduction to Signal Spaces Selin Aviyente Department of Electrical and Computer Engineering Michigan State University January 12, 2010 Motivation Outline 1 Motivation 2 Vector Space 3 Inner Product
More informationMOMENTS OF HYPERGEOMETRIC HURWITZ ZETA FUNCTIONS
MOMENTS OF HYPERGEOMETRIC HURWITZ ZETA FUNCTIONS ABDUL HASSEN AND HIEU D. NGUYEN Abstract. This paper investigates a generalization the classical Hurwitz zeta function. It is shown that many of the properties
More informationGeometric Series and the Ratio and Root Test
Geometric Series and the Ratio and Root Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 5, 2018 Outline 1 Geometric Series
More informationa 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real
More informationa 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real
More informationWeek 910: Recurrence Relations and Generating Functions
Week 910: Recurrence Relations and Generating Functions April 3, 2017 1 Some number sequences An infinite sequence (or just a sequence for short is an ordered array a 0, a 1, a 2,..., a n,... of countably
More informationIntroduction to Spherical Harmonics
Introduction to Spherical Harmonics Lawrence Liu 3 June 4 Possibly useful information. Legendre polynomials. Rodrigues formula:. Generating function: d n P n x = x n n! dx n n. wx, t = xt t = P n xt n,
More informationPhysics 70007, Fall 2009 Answers to Final Exam
Physics 70007, Fall 009 Answers to Final Exam December 17, 009 1. Quantum mechanical pictures a Demonstrate that if the commutation relation [A, B] ic is valid in any of the three Schrodinger, Heisenberg,
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georgia Tech PHYS 612 Mathematical Methods of Physics I Instructor: Predrag Cvitanović Fall semester 2012 Homework Set #5 due October 2, 2012 == show all your work for maximum credit, == put labels, title,
More information