Mathematics reference David Miller Quantum Mechanics for Scientists and Engineers On-Line Course 1 Elementary mathematical expressions Quadratic equations 2 2 a b a b a b (1) The solutions to the general quadratic equation 2 0 ax bx c (2) are 2 4 2 b b ac x a (3) Taylor and Maclaurin series (power-series expansion) The Taylor series 2 2 2 1! 2! ! n n n a a a x a x a x a df d f d f f x f a dx dx n dx (4) gives a useful way of approximating a function near to some specific point x a , giving a power-series expansion in n x a for the function near that point. The Maclaurin series 2 2 2 0 0 0 0 1! 2! ! n n n x df x d f x d f f x f dx dx n dx (5) is a special case of the Taylor series where we are expanding around the point 0 x . Power-series expansions of common functions For small a, the Maclaurin expansions of various common functions are, to first order 1 1 /2 a a (6) 1 1 1 a a (7) sin a a (8) tan a a (9) 2 cos 1 2 a a (10) exp 1 a a (11)
2 2a b a b a b (1) The solutions to the general quadratic
equation
2 0ax bx c (2) are
2 4
2b b acx
a (3)
Taylor and Maclaurin series (power-series expansion) The Taylor
series
2 2
21! 2! !
n n
na a a
x a x a x adf d f d ff x f adx dx n dx
(4)
gives a useful way of approximating a function near to some
specific point x a , giving a power-series expansion in nx a for
the function near that point. The Maclaurin series
2 2 20 0 0
01! 2! !
n n
nx df x d f x d ff x fdx dx n dx
(5)
is a special case of the Taylor series where we are expanding
around the point 0x . Power-series expansions of common
functions
For small a, the Maclaurin expansions of various common
functions are, to first order
1 1 / 2a a (6) 1 1
1a
a (7)
sin a a (8) tan a a (9)
2
cos 12aa (10)
exp 1a a (11)
Mathematics reference David Miller
Quantum Mechanics for Scientists and Engineers On-Line Course
2
Sine and cosine addition and product formulae
2 2sin cos 1 (12) sin sin cos cos sin (13) sin 2sin cos (14) cos
cos cos sin sin (15) 2 2 2 2cos 2 cos sin 2cos 1 1 2sin (16) 2 1cos
1 cos 2
2 (17)
2 1sin 1 cos 22
(18)
1cos cos cos cos2
(19)
1sin sin cos cos2
(20)
1sin cos sin sin2
(21)
cos cos 2cos cos2 2
(22)
sin sin 2sin cos2 2
(23)
cos cos 2sin sin2 2
(24)
sin sin 2cos sin2 2
(25)
Mathematics reference David Miller
Quantum Mechanics for Scientists and Engineers On-Line Course
3
Differential calculus Product rule
d dv duuv u vdx dx dx
(26) Quotient rule
2
du dvv ud u dx dxdx v v
(27)
Chain rule
d df dgf g xdx dg fx
(28)
Derivatives of elementary functions
1n nd x nxdx
(29)
exp expd ax a axdx
(30)
1lnd xdx x
(31)
sin cosd x xdx
(32)
cos sind x xdx
(33)
1
2
sin 11
d xdx x
(34)
1
2
tan 11
d xdx x
(35)
Mathematics reference David Miller
Quantum Mechanics for Scientists and Engineers On-Line Course
4
Integral calculus Integration by parts
( )b bba
a a
dg x df xf x dx f x g x g x dx
dx dx (36)
where we use the common notation
ba
h x h b h a (37) and, specifically, here
ba
f x g x f b g b f a g a (38) Some definite integrals
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