Principles of Quantum Mechanics

Appendix C

Some Mathematical Relations

C.1  SOME ALGEBRAIC RELATIONS

C.1.1  Expansions

(a ± b)² = a² ± 2ab + b²

 

(a ± b)³ = a³ ± 3a²b + 3ab² ± b³

 

(a ± b)⁴ = a⁴ ± 4a³ b + 6a²b² ± 4ab³ + b⁴

Here the binomial co-efficient is called ‘factorial r’.

C.1.2  Factors

a² – b² = (a + b)(a – b)

 

a³ ± b³ = (a ± b)(a² ∓ ab + b²)

 

aⁿ – bⁿ = (a – b)(a^{n – 1} + a^{n – 2} b + ··· + b^{n – 1})

 

aⁿ + bⁿ = (a + b)(a^{n – 1} – a^{n – 2} b + ··· + b^{n – 1}), for n odd.

C.1.3  Sum of Numbers

C.1.4  Binomial Series for x² < 1

(1 ± x)^–1 = 1 ∓ x + x² ∓ x³ + x⁴ ···

 

(1 ± x)^–2