1000 Solved Problems in Modern Physics

4 1 Mathematical Physics Γ(n)= ∫∞ 0 e−xxn−^1 dx (Re n>0) (1.11) Γ(n+1)=nΓ(n) (1.12) Ifnis a positive integer Γ(n+1)=n! (1.13) ...

1.1 Basic Concepts and Formulae 5 First few Hermite’s polynomials are: Ho(x)= 1 ,H 1 (x)= 2 x,H 2 (x)= 4 x^2 − 2 H 3 (x)= 8 x^3 ...

6 1 Mathematical Physics Recurrence formulas: xPn′(x)−Pn′− 1 (x)=nPn(x) Pn′+ 1 (x)−Pn′− 1 (x)=(2n+1)Pn(x) (1.32) Orthonormal pro ...

1.1 Basic Concepts and Formulae 7 Ln(x)=ex dn dxn (xne−x) (Rodrigue’s formula) (1.43) The first few polynomials are: Lo(x)= 1 ,L ...

8 1 Mathematical Physics Laplace transforms: Definition: A Laplace transform of the functionF(t)is ∫∞ 0 F(t)e−stdt=f(s) (1.55) T ...

1.1 Basic Concepts and Formulae 9 to have a stationary value (maximum or minimum). The integrand is taken to be a function of th ...

10 1 Mathematical Physics Poisson distribution The probability thatxevents occur in unit time when the mean rate of occurrence i ...

1.1 Basic Concepts and Formulae 11 Two limiting cases: (a)t 2 =∞;N=Noe−λt (Law of radioactivity) (1.72) This gives the number of ...

12 1 Mathematical Physics which are to be solved as ordinary algebraic equations to determine the best values ofmandC. (b) Parab ...

1.1 Basic Concepts and Formulae 13 Matrices Types of matrices and definitions Identity matrix: I 2 = ( 10 01 ) ;I 3 = ⎛ ⎝ 100 01 ...

14 1 Mathematical Physics Hermetian matrix IfA′= A, so thataij=ajifor all values ofiandj. Diagonal elements of an Hermitian matr ...

1.1 Basic Concepts and Formulae 15 Equivalence AandBare said to be equivalent (A∼B) if one can be obtained from the other by a s ...

16 1 Mathematical Physics The system of homogenous equations has non-trivial solutions if |A−λI|= ⎡ ⎢ ⎢ ⎢ ⎣ a 11 −λ a 12 ··· a 1 ...

1.1 Basic Concepts and Formulae 17 Equations which are not in the simple form (A) can be brought into that form by the following ...

18 1 Mathematical Physics Type IV equations reducible to linear form Some equations that are not linear can be reduced to the li ...

1.1 Basic Concepts and Formulae 19 First step: Write down the corresponding auxiliary equation Dn+p 1 Dn−^1 +p 2 Dn−^2 +···+pn= ...

20 1 Mathematical Physics Procedure: First step: Replace the RHS member of the given equation (I) by zero and solve the complime ...

1.2 Problems 21 Extract the square root of both members, separate the variables, and integrate again, introducing the second arb ...

22 1 Mathematical Physics 1.5 (a) If the field is centrally represented byF=f(x,y,z),r=f(r)r, then it is conservative conditione ...

1.2 Problems 23 Fig. 1.2Saw-tooth wave 1.19 Use the result of Problem 1.18 for the Fourier series for the square wave to prove t ...