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

Errata for 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables

Difference of Two Odd Powers

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

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

Volume of Rectangular Parallelepiped

Chapter $4$: Geometric Formulas: Rectangular Parallelepiped of Length $a$, Height $l$, Width $c$: $4.26$

Rectangular Parallelepiped of Length $a$, Height $l$, Width $c$

$4.26 \quad$ Volume $= a b c$

Volume of Parallelepiped

Chapter $4$: Geometric Formulas: Parallelepiped of Cross-sectional Area $A$ and Height $h$: $4.28$

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

Volume of Cylinder

Chapter $4$: Geometric Formulas: Cylinder of Cross-sectional Area $A$ and Slant Height $l$: $4.35.$ and $4.36.$

$4.35. \quad$ Volume $= A l = \dfrac {A h} {\sin \theta} = A h \csc \theta$

$4.36. \quad$ Lateral surface area $= p l = \dfrac {p h} {\sin \theta} = p h \csc \theta$

Cosine of Half Angle for Spherical Triangles

Chapter $5$: Trigonometric Functions: Relationships between Sides and Angles of a Spherical Triangle: $5.99$

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

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

Primitive of $\csch u$

Chapter $14$: Indefinite Integrals: General Rules of Integration: $14.30$:

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

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

Chapter $14$: Indefinite Integrals: Integrals involving $a x + b$: $14.73$

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

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

Chapter $14$: Indefinite Integrals: Integrals involving $a x + b$: $14.79$

$\ds \int \frac {\d x} {x^3 \paren {a x + b}^3} = \frac {a^4 x^2} {2 b^5 \paren {a x + b}^2} - \frac {4 a^3 x} {b^5 \paren {a x + b} } - \frac {\paren {a x + b}^2} {2 b^5 x^2} + \frac {6 a^2} {b^5} \ln \size {\frac x {a x + b} } + C$

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

Chapter $14$: Indefinite Integrals: Integrals involving $\sqrt {a x + b}$ and $\sqrt {p x + q}$: $14.120$

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

Primitive of $\dfrac {\d x} {p^2 + q^2 \cosh^2 a x}$

Chapter $14$: Indefinite Integrals: Integrals involving $\cosh a x$: $14.584$

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

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

Chapter $14$: Indefinite Integrals: Integrals involving Inverse Hyperbolic Functions: $14.655$

$\ds \int \frac {\map {\cosh^{-1} } {x / a} } {x^2} \rd x = \dfrac {-\map {\cosh^{-1} } {x / a} } x \mp \dfrac 1 a \map \ln {\dfrac {a + \sqrt {x^2 + a^2} } x}$ $\sqbrk {- \text { if } \map {\cosh^{-1} } {x / a} > 0, + \text { if } \map {\cosh^{-1} } {x / a} < 0}$

Leibniz's Integral Rule

Chapter $15$: Definite Integrals: Leibnitz's Rule for Differentiation of Integrals

Leibnitz's Rule for Differentiation of Integrals

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

Chapter $15$: Definite Integrals: Definite Integrals involving Trigonometric Functions: $15.34$

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

Power Series Expansion for Hyperbolic Cotangent Function

Chapter $20$: Taylor Series: Series for Hyperbolic Functions: $20.36$

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

Rodrigues' Formula for Legendre Polynomials

Chapter $25$: Legendre Functions: Legendre Polynomials: $25.2$

If $n = 0, 1, 2, \ldots$, solutions of $25.1$ are Legendre polynomials $\map {P_n} x$ given by Rodrigue's formula

Rodrigues' Formula for Hermite Polynomials

Chapter $27$: Hermite Polynomials: Hermite Polynomials: $27.2$

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.

Rodrigues' Formula for Laguerre Polynomials

Chapter $28$: Laguerre Polynomials: Laguerre Polynomials: $28.2$

If $n = 0, 1, 2, \ldots$ then solutions of Laguerre's equation are Laguerre polynomials $\map {L_n} x$ and are given by Rodrigue's formula.

Generating Function for Laguerre Polynomials

Chapter $28$: Laguerre Polynomials: Generating Function: $28.11$

$\ds \dfrac {e^{-x t / 1 - t} } {1 - t} = \sum_{n \mathop = 0}^\infty \dfrac {\map {L_n} x t^n} {n!}$

Inch: Conversion Factors

Chapter $41$: Conversion Factors: Length

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

Ton

Chapter $41$: Conversion Factors: Force

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