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

## Murray R. Spiegel: Mathematical Handbook of Formulas and Tables

Published $1968$, Schaum

ISBN 0-07-060224-7.

### Part I: Formulas

The Greek Alphabet
1. Special Constants
2. Special Products and Factors
3. The Binomial Formula and Binomial Coefficients
4. Geometric Formulas
5. Trigonometric Functions
6. Complex Numbers
7. Exponential and Logarithmic Functions
8. Hyperbolic Functions
9. Solutions of Algebraic Equations
10. Formulas from Plane Analytic Geometry
11. Special Plane Curves
12. Formulas from Solid Analytic Geometry
13. Derivatives
14. Indefinite Integrals
15. Definite Integrals
16. The Gamma Function
17. The Beta Function
18. Basic Differential Equations and Solutions
19. Series of Constants
20. Taylor Series
21. Bernoulli and Euler Numbers
22. Formulas from Vector Analysis
23. Fourier Series
24. Bessel Functions
25. Legendre Functions
26. Associated Legendre Functions
27. Hermite Polynomials
28. Laguerre Polynomials
29. Associated Laguerre Polynomials
30. Chebyshev Polynomials
31. Hypergeometric Functions
32. Laplace Transforms
33. Fourier Transforms
34. Elliptic Functions
35. Miscellaneous Special Functions
36. Inequalities
37. Partial Fraction Expansions
38. Infinite Products
39. Probability Distributions
40. Special Moments of Inertia
41. Conversion Factors

### Part II: Tables

1. Four Place Common Logarithms
2. Four Place Common Antilogarithms
3. $\operatorname{Sin} x$ ($x$ in degrees and minutes)
4. $\operatorname{Cos} x$ ($x$ in degrees and minutes)
5. $\operatorname{Tan} x$ ($x$ in degrees and minutes)
6. $\operatorname{Cot} x$ ($x$ in degrees and minutes)
7. $\operatorname{Sec} x$ ($x$ in degrees and minutes)
8. $\operatorname{Csc} x$ ($x$ in degrees and minutes)
9. Natural Trigonometric Functions (in radians)
10. $\log \sin x$ ($x$ in degrees and minutes)
11. $\log \cos x$ ($x$ in degrees and minutes)
12. $\log \tan x$ ($x$ in degrees and minutes)
13. Conversion of radians to degrees, minutes and seconds or fractions of a degree
14. Conversion of degrees, minutes and seconds to radians
15. Natural or Napierian Logarithms $\log_e x$ or $\ln x$
16. Exponential functions $e^x$
17. Exponential functions $e^{-x}$
18a. Hyperbolic functions $\sinh x$
18b. Hyperbolic functions $\cosh x$
18c. Hyperbolic functions $\tanh x$
19. Factorial $n$
20. Gamma Function
21. Binomial Coefficients
22. Squares, Cubes, Roots and Reciprocals
23. Compound Amount: $\paren {1 + r}^n$
24. Present Value of an Amount: $\paren {1 + r}^{-n}$
25. Amount of an Annuity: $\dfrac {\paren {1 + r}^n - 1} r$
26. Present Value of an Annuity: $\dfrac {1 - \paren {1 + r}^{-n}} r$
27. Bessel functions $\map {J_0} x$
28. Bessel functions $\map {J_1} x$
29. Bessel functions $\map {Y_0} x$
30. Bessel functions $\map {Y_1} x$
31. Bessel functions $\map {I_0} x$
32. Bessel functions $\map {I_1} x$
33. Bessel functions $\map {K_0} x$
34. Bessel functions $\map {K_1} x$
35. Bessel functions $\map {\operatorname{Ber} } x$
36. Bessel functions $\map {\operatorname{Bei} } x$
37. Bessel functions $\map {\operatorname{Ker} } x$
38. Bessel functions $\map {\operatorname{Kei} } x$
39. Values for Approximate Zeros of Bessel Functions
40. Exponential, Sine and Cosine Integrals
41. Legendre Polynomials $\map {P_n} x$
42. Legendre Polynomials $\map {P_n} {\cos \theta}$
43. Complete Elliptic Integrals of First and Second Kinds
44. Incomplete Elliptic Integrals of the First Kind
45. Incomplete Elliptic Integrals of the Second Kind
46. Ordinates of the Standard Normal Curve
47. Areas under the Standard Normal Curve
48. Percentile Values for Student's $t$ Distribution
49. Percentile Values for the Chi Square Distribution
50. $95$th Percentile Values for the $F$ Distribution
51. $99$th Percentile Values for the $F$ Distribution
52. Random Numbers
Index of Special Symbols and Notations
Index

## Errata

Chapter $2$: Special Products and Factors: $2.22$: Difference of Two Odd Powers:

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

Chapter $5$: Trigonometric Functions: Relationships between Sides and Angles of a Spherical Triangle: $5.99$: Cosine of Half Angle for Spherical Triangles:

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

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

Chapter $14$: Indefinite Integrals: General Rules of Integration: $14.30$: Primitive of $\csch u$:

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

Chapter $14$: Indefinite Integrals: Integrals involving $a x + b$: $14.73$: Primitive of $\dfrac 1 {\paren {a x + b}^3}$:

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

Chapter $14$: Indefinite Integrals: Integrals involving $a x + b$: $14.79$: Primitive of $\dfrac 1 {x^3 \paren {a x + b}^3}$:

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

Chapter $14$: Indefinite Integrals: Integrals involving $\sqrt {a x + b}$ and $\sqrt {p x + q}$: $14.120$: Primitive of $\dfrac 1 {\sqrt {\paren {a x + b} \paren {p x + q} } }$:

$\displaystyle \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \begin {cases} \dfrac 2 {\sqrt {a p} } \map \ln {\sqrt {p \paren {a x + b} } + \sqrt {a \paren {p x + q} } } + C & : \dfrac {b p - a q} p > 0 \\ \dfrac 2 {-\sqrt {a p} } \arctan \sqrt {\dfrac {-p \paren {a x + b} } {a \paren {p x + q} } } + C & :\dfrac {b p - a q} p < 0 \\ \end{cases}$

Chapter $14$: Indefinite Integrals: Integrals involving $\sqrt {a^2 - x^2}$: $14.246$: Primitive of $x^2 \sqrt {a^2 - x^2}$:

$\displaystyle \int x^2 \sqrt {a^2 - x^2} \rd x = \frac {-x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {a^2 x \sqrt {a^2 - x^2} } 8 + \frac {a^4} 8 \arcsin \frac x a + C$