2.3 Series, summations, and progressions

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2.3 Series, summations, and progressions	27

2.3Series, summations, and progressions

Progressions and summations

Sn = a+ (a+ d) + (a+ 2d) + · · ·

number of terms

+ [a+ (n− 1)d]

(2.102)

sum of n successive

Arithmetic

terms

progression

[2a+ (n− 1)d]

(2.103)

ﬁrst term

common di erence

(a+ l)

(2.104)

last term

Sn = a+ ar + ar2 +

· · ·

+ arn−1

(2.105)

Geometric

= a

1 − rn

(2.106)

common ratio

progression

−

S∞ =

(|r| < 1)

(2.107)

1 − r

Arithmetic

(x1 + x2 + · · · + xn)

mean

x a =

(2.108)

arithmetic mean

Geometric

x g = (x1x2x3 ...xn)

1/n

(2.109)

geometric mean

mean

x h = n

−

Harmonic mean

+ · · · +

(2.110)

harmonic mean

Relative mean

x a ≥ x g ≥ x h

if xi > 0 for all i

(2.111)

magnitudes

i =

(n+ 1)

(2.112)

i=1

i2 =

(n+ 1)(2n+ 1)

(2.113)

i=1

i3 =

(n+ 1)2

(2.114)

i=1

Summation

+ 3n− 1)

formulas

(n+ 1)(2n+ 1)(3n

(2.115)

dummy integer

i=1

∞

(−1)i+1

= 1

−

+ ... = ln2

(2.116)

i=1

−

∞

(−1)i+1

= 1

−

+ ... =

(2.117)

i=1 2i

3 5

π2

∞

1 1

= 1 +

+ ... =

(2.118)

i=1

Euler’s

lim

lnn

(2.119)

Euler’s constant

constanta

= n→∞ 1 + 2

· · ·

n −

aγ 0.577215664...

28 Mathematics

Power series

Binomial

(1 + x)n = 1 + nx+

n(n− 1)

n(n− 1)(n− 2)

· · ·

(2.120)

seriesa

Binomial

Cr ≡ r ≡

(2.121)

coe cientb

r!(n− r)!

Binomial

theorem

(a+ b)

= k=0

k a − b

(2.122)

Taylor series

(1)

(2)

xn−1

(about a)c

f(a+ x) = f(a) + xf

(a) +

(a) + · · · +

−

(a) + · · ·

(2.123)

−

1)!

Taylor series

f(a + x) = f(a) + (x

a +

(x · )2

a +

(x · )3

· · ·

(2.124)

(3-D)

Maclaurin

(1)

(2)

xn−1

series

f(x) = f(0) + xf

(0) +

(0) + · · ·

−

(0) + · · ·

(2.125)

−

1)!

aIf n is a positive integer the series terminates and is valid for all x. Otherwise the (inﬁnite) series is convergent for

|x| < 1.

bThe coe cient of xr in the binomial series.

cxf(n)(a) is x times the nth derivative of the function f(x) with respect to x evaluated at a, taken as well behaved around a. (x · )nf|a is its extension to three dimensions.

Limits

ncxn → 0

n → ∞

if |x| < 1 (for any ﬁxed c)

(2.126)

xn/n! → 0 as

n → ∞ (for any ﬁxed x)

(2.127)

(1 + x/n)n → ex

n → ∞

(2.128)

xlnx → 0

as x → 0

(2.129)

sinx

→ 1

x → 0

(2.130)

∞

f(x)

f(1)(a)

(l’Hopital’sˆ rule)

(2.131)

f(a) = g(a) = 0 or

then

lim

g(1)(a)

x→a g(x)

2.3 Series, summations, and progressions	29

Series expansions

exp(x)

(for all x)

1 + x+ 2! + 3! + · · ·

(2.132)

ln(1 + x)

x−

−

+ · · ·

(2.133)

(−1 < x ≤ 1)

1 + x

+ · · ·

(2.134)

(|x| < 1)

1 − x

cos(x)

(for all x)

1 − 2! +

4! −

6! + · · ·

(2.135)

sin(x)

(for all x)

x− 3! +

5! −

7! + · · ·

(2.136)

tan(x)

2x5

17x7

· · ·

(2.137)

(|x| < π/2)

315

sec(x)

5x4

61x6

1 +

+ · · ·

(2.138)

(|x| < π/2)

720

csc(x)

7x3

31x5

+ · · ·

(|x| < π)

(2.139)

360

15120

cot(x)

2x5

−

− · · ·

(2.140)

(|x| < π)

945

arcsin(x)a

1 · 3

1 · 3 · 5

(2.141)

( x

< 1)

2 3 2

4 5 2

6 7 · · ·

| |

x−

−

+ · · ·

(|x| ≤ 1)

arctan(x)

(2.142)

(x > 1)

−

· · ·

1 + 13

15 +

(x < −1)

−

· · ·

cosh(x)

(for all x)

1 + 2! +

4! +

6! + · · ·

(2.143)

sinh(x)

(for all x)

x+ 3! +

5! +

7! + · · ·

(2.144)

tanh(x)

2x5

17x7

x−

−

+ · · ·

(2.145)

(|x| < π/2)

315

aarccos(x) = π/2 − arcsin(x). Note that arcsin(x) ≡ sin−1(x) etc. barccot(x) = π/2 − arctan(x).

30 Mathematics

Inequalities

Triangle

|a1| − |a2| ≤ |a1 + a2| ≤ |a1| + |a2| ;

(2.146)

inequality

|ai|

(2.147)

≤

i=1

≥

...

≥

(2.148)

≥

Chebyshev

and

b1 ≥ b2 ≥ b3

≥ ... ≥ bn

(2.149)

inequality

then

aibi ≥

ai n

(2.150)

i=1

Cauchy

inequality

aibi ≤

ai2

bi2

(2.151)

i=1

inequality

f(x)g(x) dx

≤ a

[f(x)]2 dx a

[g(x)]2 dx

(2.152)

Schwarz

2.4 Complex variables Complex numbers

complex variable

Cartesian form

z = x+ iy

(2.153)

= −1

x,y

real variables

amplitude (real)

Polar form

z = reiθ = r(cosθ + isinθ)

(2.154)

phase (real)

Modulusa

|z| = r = (x2 + y2)1/2

(2.155)

modulus of z

|z1 · z2| = |z1| · |z2|

(2.156)

θ = argz = arctan

(2.157)

Argument

argz

argument of z

arg(z1z2) = argz1 + argz2

(2.158)

Complex

z = x

−

iy = re−iθ

(2.159)

arg(z ) = − argz

(2.160)

complex conjugate of

conjugate

z = reiθ

z · z = |z|2

(2.161)

Logarithmb

lnz = lnr + i(θ + 2πn)

(2.162)

integer

aOr “magnitude.”

bThe principal value of lnz is given by n = 0 and −π < θ ≤ π.

2.4 Complex variables	31

Complex analysisa

f(z) = u(x,y) + iv(x,y)

Cauchy–

then

∂u

∂v

(2.163)

Riemann

∂x

∂y

equationsb

∂u

∂v

= −

(2.164)

∂y

∂x

Cauchy–

c f(z) dz = 0

Goursat

(2.165)

theoremc

Cauchy

f(z0) =

f(z)

(2.166)

integral

2πi c z − z0

formulad

f(n)(z0) =

f(z)

(2.167)

2πi

(z − z0)n+1

f(z) =

∞

an(z − z0)n

(2.168)

Laurent

n=−∞

expansion

f(z )

where

an =

(2.169)

2πi

(z − z0)n+1

Residue

c f(z) dz = 2πi enclosed residues

(2.170)

theorem

z	complex variable
i	i2 = −1	2
x,y	real variables	2
f(z)	function of z
u,v	real functions

(n)	nth derivative
	nth derivative
an	Laurent coe cients
a−1	residue of f(z) at z0
z	dummy variable

cz0

aClosed contour integrals are taken in the counterclockwise sense, once. bNecessary condition for f(z) to be analytic at a given point.

cIf f(z) is analytic within and on a simple closed curve c. Sometimes called “Cauchy’s theorem.” dIf f(z) is analytic within and on a simple closed curve c, encircling z0.

eOf f(z), (analytic) in the annular region between concentric circles, c1 and c2, centred on z0. c is any closed curve in this region encircling z0.

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