Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Third Edition/Errata

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Errata for 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.)

Difference of Two Odd Powers

Chapter $2$: Special Products and Factors: $2.22.$

$x^{2 n} - y^{2 n} = \paren {x - y} \paren {x + y} \paren {x^{n - 1} + x^{n - 2} y + x^{n - 3} y^2 + \dotsb} \paren {x^{n - 1} - x^{n - 2} y + x^{n - 3} y^2 - \dotsb}$


Inch: Conversion Factors

Chapter $6$: Conversion Factors: Length

$1$ inch (in.) $= 2.540$ cm


Mil: Conversion Factors

Chapter $6$: Conversion Factors: Length

$1$ millimeter $= 10^{-3}$ in


Ton

Chapter $6$: Conversion Factors: Force

$1$ U.S. short ton $2000 \, \mathrm {lbwt}$; $\quad 1$ long ton $2240 \, \mathrm {lbwt}$; $\quad 1$ metric ton $2205 \, \mathrm {lbwt}$


Volume of Parallelepiped

Chapter $7$: Geometric Formulas: Parallelepiped of Cross-sectional Area $A$ and Height $h$: $7.28.$

Volume $= A h = a b c \sin \theta$


Volume of Cylinder

Chapter $7$: Geometric Formulas: Cylinder of Cross-sectional Area $A$ and Slant Height $l$: $7.35.$ and $7.36.$

$7.35. \quad$ Volume $= A h = A l \sin \theta$
$7.36. \quad$ Lateral surface area $= p h = p l \sin \theta$


Cosine of Half Angle for Spherical Triangles

Chapter $12$: Trigonometric Functions: Relationships between Sides and Angles of a Spherical Triangle: $12.99.$

$\cos \dfrac A 2 = \sqrt {\dfrac {\sin s \, \map \sin {s - c} } {\sin b \sin c} }$

where $s = \dfrac {a + b + c} 2$.


Primitive of $\csch u$

Chapter $16$: Indefinite Integrals: General Rules of Integration: $16.30.$

$\ds \int \csch u \rd u = -\map {\coth^{-1} } {e^u} + C$


Bernoulli Numbers

Chapter $17$: Tables of Special Indefinite Integrals

Some integrals contain the Bernouilli numbers $B_n$ and the Euler numbers $E_n$ defined in Chapter $23$.


Primitive of $\dfrac 1 {\paren {a x + b}^3}$

Chapter $17$: Tables of Special Indefinite Integrals: $(1)$ Integrals Involving $a x + b$: $17.1.11.$

$\ds \int \frac {\d x} {\paren {a x + b}^3} = \frac {-1} {2 \paren {a x + b}^2} + C$


Primitive of $\dfrac 1 {\sqrt {\paren {a x + b} \paren {p x + q} } }$

Chapter $17$: Tables of Special Indefinite Integrals: $(5)$ Integrals Involving $\sqrt {a x + b}$ and $\sqrt {p x + q}$: $17.5.1.$

$\ds \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \begin {cases} \dfrac 2 {\sqrt {a p} } \map \ln {\sqrt {p \paren {a x + b} } + \sqrt {a \paren {p x + q} } } \\ \\ \dfrac 2 {\sqrt {-a p} } \tan^{-1} \sqrt {\dfrac {-p \paren {a x + b} } {a \paren {p x + q} } } \end{cases}$


Integrals Involving $x^2 - a^2$, $x^2 < a^2$

Chapter $17$: Tables of Special Indefinite Integrals: $(8)$ Integrals Involving $x^2 - a^2$, $x^2 < a^2$

$(8)$ Integrals Involving $x^2 - a^2$, $x^2 < a^2$


Primitive of $\dfrac 1 {x^2 \paren {x^4 - a^4} }$

Chapter $17$: Tables of Special Indefinite Integrals: $(15)$ Integrals Involving $x^4 \pm a^4$: $17.15.13.$

$\ds \int \frac {\d x} {x^3 \paren {x^4 - a^4} } = \frac 1 {a^4 x} + \frac 1 {4 a^5} \ln \size {\frac {x - a} {x + a} } + \frac 1 {2 a^5} \arctan \frac x a + C$


Leibniz's Integral Rule

Chapter $18$: Definite Integrals: Leibnitz's Rules for Differentiation of Integrals

Leibnitz's Rules for Differentiation of Integrals


Definite integral $\ds \int_0^\infty \frac {\sin p x \cos q x} x \rd x$

Chapter $18$: Definite Integrals: Definite Integrals involving Trigonometric Functions: $18.34.$

$\ds \int_0^\infty \frac {\sin p x \cos q x} x \rd x = \begin {cases} 0 & : p > q > 0 \\ \\ \dfrac \pi 2 & : 0 < p < q \\ \\ \dfrac \pi 4 & : p = q > 0 \end {cases}$


Power Series Expansion for Hyperbolic Cotangent Function

Chapter $22$: Taylor Series: Series for Hyperbolic Functions: $22.36.$

$\coth x = \dfrac 1 x + \dfrac x 3 - \dfrac {x^3} {45} + \dfrac {2 x^5} {945} + \cdots \dfrac {\paren {-1}^n 2^{2 n} B_n x^{2 n - 1} } {\paren {2 n}!} + \cdots \quad 0 < \size x < \pi$


Sum of Reciprocals of Even Powers of Integers Alternating in Sign: Corollary

Chapter $23$: Bernoulli and Euler Numbers: Series involving Bernoulli and Euler Numbers: $23.10.$

\(\ds B_n\) \(=\) \(\ds \dfrac {2 \paren {2 n}!} {\paren {2^{2 n - 1} - 1} \pi^{2 n} } \set {1 - \dfrac 1 {2^{2 n} } + \dfrac 1 {3^{2 n} } - \cdots}\)


Legendre Polynomial Derivative Recursion Formula: Corollary 3

Chapter $28$: Legendre and Associated Legendre Functions: Recurrence Formulas for Legendre Polynomials: $28.24.$

$\paren {x^2 - 1} \dfrac \d {\d x} \map {P_n} x - n x \map {P_n} x - n \map {P_{n - 1} } x$


Rodrigues' Formula for Hermite Polynomials

Chapter $29$: Hermite Polynomials: Hermite Polynomials: $29.2.$

If $n = 0, 1, 2, \ldots$, then a solution of Hermite's equation is the Hermite polynomial $\map {H_n} x$ given by Rodrigue's formula.


Laplace Transform of $t \cosh a t$

Chapter $\S 33$: Laplace Transforms: Table of Special Laplace Transforms: $33.51.$

$\begin{array}{c|c} \map f s & \map F t \\ \hline \dfrac {s^2} {\paren {s^2 - a^2}^{3/2} } & t \cosh a t \\ \end{array}$


Hölder's Inequality

Chapter $37$: Inequalities: Holder's Inequality

Holder's Inequality
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