The pur-pose of this handbook is to supply a collection of mathematical formulas and tables which will prove to be valuable to students and research workers in. Speigel, M.R., Mathematical Handbook of Formulas and Tables. (Schaum's Outline Series, McGraw-Hill, ). Physical Constants. Based on the “Review of . SCHAUM'S outlines Mathematical Handbook of Formulas and Tables This page The material in this eBook also appears in the print version of.
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Schaum's Outline Series Mathematica, a computer system designed to perform complex mathematical calculus, differential equations, and linear algebra. that you consider downloading a mathematical handbook (i.e. Schaum's Outline Series: Mathematical Handbook of Formulas and Tables by Murray R. Spiegel. Schaum's Outline of Mathematical Handbook of Formulas and Tables, 4th Outline . Taschenbuch der Mathematik, desktop edition (de)(s).pdf" (М).
Similarly the derivatives in 14 are denoted by fxx, fxv, fyx, fm respectively.
The differential of f x, y is defined as 15 where Generalizations of these results are easily made. Similarly the numerator for y is found by replacing the second column of the denominator by ci, c2. This procedure is often called Cramer's rule. In case the denominator in 19 is zero, the two lines represented by 17 do not meet in one point but are either coincident or parallel. The ideas are easily extended. Thus consider the equations 21 representing 3 planes. The general theory of determinants, of which the above results are special cases, is considered in Chapter Thus possible points at which f x,y has relative maxima or minima are obtained by solving simultaneously the equations 26 Extensions to functions of more than two variables are similar.
Generalizations can be made [see Problems 1. Because several ideas involved in the theory may be new to some students, we postpone consideration of this topic to Chapter 6.
We define operations with complex numbers as follows. The commutative, associative and distributive laws of page 1 also apply to complex numbers. Two complex numbers are equal if and only if their real and imaginary parts are respectively equal. The angle called the amplitude or argument of z abbreviated argz.
We can use this to determine roots of complex numbers. Many of the ideas presented in this chapter involving real numbers can be extended to complex numbers. The magnitude of the vector is determined by the length of the arrow, using an appropriate unit. Notation for Vectors A vector is denoted by a bold faced letter such as A Fig. The magnitude is denoted by A or A. The tail end of the arrow is called the initial point, while the head is called the terminal point. Fundamental Definitions 1.
Equality of vectors. Two vectors are equal if they have the same magnitude and direction. Multiplication of a vector by a scalar. Sums of vectors. This definition is equivalent to the parallelogram law for vector addition as indicated in Fig.
Thus, Fig. Unit vectors. A unit vector is a vector with unit magnitude. If i, j, k are unit vectors in the directions of the positive x, y, z axes, then k A A3k Fundamental results follow: Cross or Vector Product We assume that all derivatives exist unless otherwise specified. Formulas Involving Derivatives d dB dA Divide the curve into n parts by P2 points of subdivision x1, y1, z1 ,.
The result The line integral Properties of Line Integrals p2 P1 In such a case, C P2 Subdivide the region into n parts by lines parallel to the x and y axes as indicated. In such a case, the integral can also be written as b f2 x Fig. The result can also be written as These are called double integrals or area integrals. The ideas can be similarly extended to triple or volume integrals or to higher multiple integrals.
Surface Integrals z Subdivide the surface S see Fig. Then S the surface integral of the normal component of A over S is defined as n Then see Fig.
Then Similarly, we define the u2 and u3 coordinate curves through P. If e1, e2, e3 are mutually perpendicular, the curvilinear coordinate system is called orthogonal.
If dV is the element of volume, then Transformation of Multiple Integrals Result Cylindrical coordinates. Spherical coordinates. Spherical Coordinates r, q, f See Fig. They are confocal parabolas Fig. Elliptic Cylindrical Coordinates u, y, z They are confocal ellipses and hyperbolas.
Elliptic cylindrical coordinates. The third set of coordinate surfaces consists of planes passing through this axis. Oblate Spheroidal Coordinates x, h, f The third set of coordinate surfaces are planes passing through this axis.
Bipolar coordinates. Some special cases are The value x, which may be different in the two forms, lies between a and x. The result holds if f x has continuous derivatives of order n at least. These series, often called power series, generally converge for all values of x in some interval called the interval of convergence and diverge for all x outside this interval. Complex Form of Fourier Series Assuming that the series Some Important Results Then the following series expansions hold under the conditions indicated.
This is called the addition formula for Bessel functions. Orthogonal Series of Legendre Polynomials We restrict ourselves to the important case where m, n are nonnegative integers. We have Orthogonal Series The functions Qnm x satisfy the same recurrence relations as Pnm x see Recurrence Formulas Chebyshev Polynomials of the First Kind A solution of Special Cases Complex Inversion Formula The inverse Laplace transform of f s can be found directly by methods of complex variable theory.
Section X: We distinguish these two cases using different notation as follows: First we give formulas for the data coming from a sample. This is followed by formulas for the population. Grouped Data Frequently, the sample data are collected into groups grouped data. A group refers to a set of numbers all with the same value xi, or a set class of numbers in a given interval with class value xi. In such a case, we assume there are k groups with fi denoting the number of elements in the group with value or class value xi.
Thus, the total number of data items is Accordingly, some of the formulas will be designated as a or as b , where a indicates ungrouped data and b indicates grouped data. That is: Sample mean: Mode The mode is the value or values which occur most often. Suppose that there are k sample sets and that each sample set has ni elements and a mean x. Midrange The midrange is the average of the smallest value x1 and the largest value xn.
Population mean: Observe that the formula for the population mean m is the same as the formula for the sample mean x. On the other hand, the formula for the population standard deviation s is not the same as the formula for the sample standard deviation s.
This is the main reason we give separate formulas for m and x. Sample variance: Sample standard deviation: Consider the following frequency distribution: Hence, by Here M. Sample range: There are three quartiles: Five-number summary: Innerquartile range: Semi-innerquartile range: The rth moment: The rth moment about the mean x: The rth absolute moment about mean x: Coefficient of skewness: Momental skewness: Coefficient of kurtosis: Coefficient of excess kurtosis: Quartile coefficient of skewness: Population variance: Population standard deviation: The primary objective is to determine whether there is a mathematical relationship, such as a linear relationship, between the data.
The scatterplot of the data is simply a picture of the pairs of values as points in a coordinate plane.
Sample correlation coefficient: An alternative formula for computing r follows: The sample covariance of x and y is denoted and defined as follows: Sample covariance: Consider the following data: The correlation coefficient r for the data may be obtained by first constructing the table in Fig. Then, by Formula The squares error between the line L and the data points is defined by It can be shown that such a line L exists and is unique.
Suppose we want the line L of best fit for the data in Example Using the table in Fig. Three such types of curve are discussed as follows. Polynomial function of degree m: Then log a and log b are obtained from transformed data points.
Consider the following data which indicates exponential growth: The data points and C are depicted in Fig. The log a and b are obtained from transformed data points.
It would be convenient if all subsets of S could be events. Unfortunately, this may lead to contradictions when a probability function is defined on the events. Thus, the events are defined to be a limited collection C of subsets of S as follows. That is, C has the following three properties: Thus, the events form a collection that is closed under unions, intersections, and complements of denumerable sequences. However, if S is nondenumerable, then only certain subsets of S can be the events.
If Condition ii in Definition First, for completeness, we list basic properties of the set operations of union, intersection, and complement. Sets satisfy the properties in Table The following are equivalent: Recall that the union and intersection of any collection of sets is defined as follows: Let P be a real-valued function defined on the class C of events of a sample space S.
Then P is called a probability function, and P A is called the probability of an event A, when the following axioms hold: Axiom [P3] implies an analogous axiom for any finite number of sets. The following properties follow directly from the above axioms.
Limits of Sequences of Events Note lim An exists when the sequence is monotonic. Borel-Cantelli Lemma Suppose Aj is any sequence of events in a probability space. Extension Theorem Let F be a field of subsets of S. The conditional probability of an event A given E is denoted and defined as follows: This theorem can be genealized as follows: A lot contains 12 items of which 4 are defective.
Three items are drawn at random from the lot one after the other. Find the probabiliy that all three are nondefective.
A convenient way of describing such a process is by means of a probability tree diagram, illustrated below, where the multiplication theorem A coin is selected at random and is tossed. An item is ran- domly selected. Similarly, P D 0. We find P D by adding the three probability paths to D: That is, events A and B are independent if the occurrence of one of them does not influence the occur- rence of the other.
Consider the following events for a family with children where we assume the sample space S is an equiprobable space: Hence, E and F are independent.
Hence, E and F are dependent. The concept of independent repeated trials, when S is a finite set, is formalized as follows. Let S be a finite probability space. They race three times. Each element in B has the same probability 0. A random variable X on the sample space S is a function from S into the set R of real numbers such that the preimage of every interval of R is an event of S.
If S is a discrete sample space in which every subset of S is an event, then every real-valued function on S is a random variable. On the other hand, if S is uncountable, then certain real-valued functions on S may not be random variables. Let X and Y be random variables on the same sample space S. The following short notation is used: Then X induces a function f x on RX as follows: The pair xi, f xi , usually given by a table, is called the probability distribution or probability mass function of X.
Suppose X has the following probability distribution: Let X be the continuous random variable with the following density function: The function F x has the following properties: If X is the discrete random variable with distribution f x , then F x is the following step function: That is, d Consider the random variable X in Example Binomial Distribution: Poisson Distribution: Hypergeometric Distribution: Normal Distribution: F Distribution: Gauss-Legendre formula in interval —1, 1 1 n Nonlinear equation: Fixed point nonlinear equation: Bisection Method The following theorem applies: Intermediate Value Theorem: Bisection method: Initial step: Repetitive step: Fixed-Point Iteration The following definition and theorem apply: Fixed-point theorem: Suppose that g is a contraction mapping on a, b.
Then g has a unique fixed point in a, b. Given such a contraction mapping g, the following method may be used. Fixed-point iteration First-Order Methods Forward Euler method first-order explicit method Second-order difference approximation Computational boundary condition A second-order finite-difference approximation Jacobi method 1 k We assume the coefficient matrix A is partitioned as follows: Richardson method The process is often called an annuity.
Cases where a symbol has more than one meaning will be clear from the context. Associated Laguerre polynomials, binomial, 7 See also Laguerre polynomials multinomial, 9 Associated Legendre functions, Complementary error function, See also Legendre functions Complex: Distributions, probability, mean G. Logarithms, 53—55 definite see Definite integrals of complex numbers, 55 improper, Griggsian, 53 indefinite see Indefinite integrals line, Maclaurin series, multiple, Mean, surface, continuous random variable, Integration, 64 See also Integrals deviation M.
Normal curve, — curve, distribution, line, Normal equations for least-squares line, Legendre functions, — Null function, of the second kind, Numbers: