# 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

Save this PDF as:

Size: px
Start display at page:

Download "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"

## 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 spherical-polar 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 square-integrable 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 well-suited(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 r-s 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 non-integer 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 Gram-Schmidt 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 square-integrable. If the function has norm, then it is normalized. Any square-integrable 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 square-integrable 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 Gram-Schmidt 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 non-orthogonal 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 non-negative 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

### Math221: 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

### d 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

### S. 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

### Two special equations: Bessel s and Legendre s equations. p Fourier-Bessel and Fourier-Legendre series. p

LECTURE 1 Table of Contents Two special equations: Bessel s and Legendre s equations. p. 259-268. Fourier-Bessel and Fourier-Legendre series. p. 453-460. Boundary value problems in other coordinate system.

### ENGI 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

### Mathematics 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 1-1 0 n 3. A square matrix

### 4 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.

### Mathematics 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

### swapneel/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 =

### MATH3383. 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

### Chapter 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

### The 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

### Relevant 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,

### Solutions: Problem Set 3 Math 201B, Winter 2007

Solutions: Problem Set 3 Math 201B, Winter 2007 Problem 1. Prove that an infinite-dimensional Hilbert space is a separable metric space if and only if it has a countable orthonormal basis. Solution. If

### Bessel 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

### ODE 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;

### Finite-dimensional spaces. C n is the space of n-tuples 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 pre-hilbert space, or a unitary space) if there is a mapping (, )

### Final 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.

### Series 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.

### 5.4 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

### Differential Equations

Differential Equations Problem Sheet 1 3 rd November 2011 First-Order Ordinary Differential Equations 1. Find the general solutions of the following separable differential equations. Which equations are

### Lecture 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

### LEGENDRE 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

### Advanced 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

### Laplace 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

### Inner 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)

### Mathematics 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

### PHYS 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,

### Infinite 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

### Series 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

### Hilbert Space Problems

Hilbert Space Problems Prescribed books for problems. ) Hilbert Spaces, Wavelets, Generalized Functions and Modern Quantum Mechanics by Willi-Hans Steeb Kluwer Academic Publishers, 998 ISBN -7923-523-9

### Collection of formulae Quantum mechanics. Basic Formulas Division of Material Science Hans Weber. Operators

Basic Formulas 17-1-1 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

### Notes on Special Functions

Spring 25 1 Notes on Special Functions Francis J. Narcowich Department of Mathematics Texas A&M University College Station, TX 77843-3368 Introduction These notes are for our classes on special functions.

### which 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: Gram-Schmidt 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.

### Quantum 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

### Spherical Harmonics on S 2

7 August 00 1 Spherical Harmonics on 1 The Laplace-Beltrami 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

### Bessel 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

### Hilbert 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,

### On 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

### Physics 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

### Power 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 non-negative constant, i.e α 0, is called

### Vector 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 +

### Electromagnetism HW 1 math review

Electromagnetism HW math review Problems -5 due Mon 7th Sep, 6- due Mon 4th Sep Exercise. The Levi-Civita symbol, ɛ ijk, also known as the completely antisymmetric rank-3 tensor, has the following properties:

### Background 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

### Scientific 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

### Review 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

### Part 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

### Some 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)

### Math 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..

### Physics 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

### LECTURE 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

### Asymptotics of Integrals of. Hermite Polynomials

Applied Mathematical Sciences, Vol. 4, 010, no. 61, 04-056 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

### Quantum Mechanics-I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras. Lecture - 21 Square-Integrable Functions

Quantum Mechanics-I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 21 Square-Integrable Functions (Refer Slide Time: 00:06) (Refer Slide Time: 00:14) We

### Introduction to Sturm-Liouville Theory and the Theory of Generalized Fourier Series

CHAPTER 5 Introduction to Sturm-Liouville Theory and the Theory of Generalized Fourier Series We start with some introductory examples. 5.. Cauchy s equation The homogeneous Euler-Cauchy equation (Leonhard

### Problem 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

### Legendre 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

### Power 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

### Qualification Exam: Mathematical Methods

Qualification Exam: Mathematical Methods Name:, QEID#41534189: August, 218 Qualification Exam QEID#41534189 2 1 Mathematical Methods I Problem 1. ID:MM-1-2 Solve the differential equation dy + y = sin

### Page 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 One-Dimensional Simple Harmonic Oscillator: Coordinate Basis Page

### Connection 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[math-ph], J. Phys. A in 48 (015) 11504, with S. Odake nd Numazu

### Math 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,

### Chapter 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

### ELEMENTARY 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

### Nonlinear Integral Equation Formulation of Orthogonal Polynomials

Nonlinear Integral Equation Formulation of Orthogonal Polynomials Eli Ben-Naim Theory Division, Los Alamos National Laboratory with: Carl Bender (Washington University, St. Louis) C.M. Bender and E. Ben-Naim,

### ELEMENTARY 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

### CYK\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.

### Taylor 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.

### Lecture 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

### (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

### Problem 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)

### Linearization 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

### 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.

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

### APPENDIX B GRAM-SCHMIDT PROCEDURE OF ORTHOGONALIZATION. Let V be a finite dimensional inner product space spanned by basis vector functions

301 APPENDIX B GRAM-SCHMIDT 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 Gram-Schmidt Process an

### Classnotes - MA Series and Matrices

Classnotes - MA-2 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

### Second-order ordinary differential equations Special functions, Sturm-Liouville theory and transforms R.S. Johnson

Second-order ordinary differential equations Special functions, Sturm-Liouville theory and transforms R.S. Johnson R.S. Johnson Second-order ordinary differential equations Special functions, Sturm-Liouville

### 1 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

### Sample 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.)

### ELEMENTARY 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,

### Notes 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

### A 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

### Higher-order ordinary differential equations

Higher-order 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

### which implies that we can take solutions which are simultaneous eigen functions of

Module 1 : Quantum Mechanics Chapter 6 : Quantum mechanics in 3-D Quantum mechanics in 3-D For most physical systems, the dynamics is in 3-D. The solutions to the general 3-d problem are quite complicated,

### 1 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

### MAT Linear Algebra Collection of sample exams

MAT 342 - Linear Algebra Collection of sample exams A-x. (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

### Vectors in Function Spaces

Jim Lambers MAT 66 Spring Semester 15-16 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

### Special 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

### Introduction 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)

### ORTHOGONAL 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 :=

### Math 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

### Recall 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

### Introduction 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

### MOMENTS 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

### Geometric 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

### a 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

### a 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

### Week 9-10: Recurrence Relations and Generating Functions

Week 9-10: 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

### Introduction 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,

### Physics 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,

### Georgia 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,