Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.10 HYPERGEOMETRIC FUNCTIONS F(a, b, b;x)=(1−x)−a, F(^12 ,^12 ,^32 ;x^2 )=x−^1 sin−^1 x, F(1, 1 ,2;−x)=x−^1 ln(1 +x), F(^12 , ...

SPECIAL FUNCTIONS where in the second equality we have used the expression (18.165) relating the gamma and beta functions. Using ...

18.11 CONFLUENT HYPERGEOMETRIC FUNCTIONS 18.11 Confluent hypergeometric functions The confluent hypergeometric equation has the ...

SPECIAL FUNCTIONS second solution to (18.147) as a linear combination of (18.148) and (18.149) given by U(a, c;x)≡ π sinπc [ M(a ...

18.12 THE GAMMA FUNCTION AND RELATED FUNCTIONS which converges providedc>a>0.
Prove the result (18.150). SinceF(a, b, c;x ...

SPECIAL FUNCTIONS 18.12.1 The gamma function Thegamma functionΓ(n) is defined by Γ(n)= ∫∞ 0 xn−^1 e−xdx, (18.153) which converge ...

18.12 THE GAMMA FUNCTION AND RELATED FUNCTIONS Γ( ) 1 2 2 3 4 4 − (^2) − 1 − 2 − 4 − 3 − 4 − 6 6 n n Figure 18.9 The gamma funct ...

SPECIAL FUNCTIONS If we letx=n+y,then lnx=lnn+ln ( 1+ y n ) =lnn+ y n − y^2 2 n^2 + y^3 3 n^3 −···. Substituting this result int ...

18.12 THE GAMMA FUNCTION AND RELATED FUNCTIONS Changing variables to plane polar coordinates (ρ, φ)givenbyx=ρcosφ,y=ρsinφ,we obt ...

SPECIAL FUNCTIONS which is the required result. We note that is it conventional to define, in addition, the functions P(a, x)≡ ...

18.13 EXERCISES Y 00 = √ 1 4 π,Y 0 1 = √ 3 4 πcosθ, Y 1 ±^1 =∓ √ 3 8 πsinθexp(±iφ),Y 0 2 = √ 5 16 π(3 cos (^2) θ−1), Y 2 ±^1 =∓ ...

SPECIAL FUNCTIONS and hence that theHn(x) satisfy the Hermite equation y′′− 2 xy′+2ny=0, wherenis an integer≥0. Use Φ to prove t ...

18.13 EXERCISES [ You will find it convenient to use ∫∞ −∞ x^2 ne−x 2 dx= (2n)! √ π 22 nn! for integern≥0. ] 18.9 By initially w ...

SPECIAL FUNCTIONS Deduce the value of J= ∫∞ 0 (u+2)^2 (u^2 +4)^5 /^2 du. 18.15 The complex functionz! is defined by z!= ∫∞ 0 uze ...

18.13 EXERCISES (a) use their series representation to prove that b d dz M(a, c;z)=aM(a+1,c+1;z); (b) use an integral representa ...

SPECIAL FUNCTIONS 18.24 The solutionsy(x, a) of the equation d^2 y dx^2 −(^14 x^2 +a)y=0 (∗) are known as parabolic cylinder fun ...

18.14 HINTS AND ANSWERS 18.15 (a) Show that the ratio of two definitions based onmandn,withm>n>−Rez, is unity, independent ...

19 Quantum operators Although the previous chapter was principally concerned with the use of linear operators and their eigenfun ...

19.1 OPERATOR FORMALISM represent different states, is a ket that represents a continuum of different states as the complex numb ...

QUANTUM OPERATORS is to produce a scalar multiple of that ket, i.e. A|ψ〉=λ|ψ〉, (19.3) then, just as for matrices and differentia ...