Schaums mathematical handbook of formulas and tables pdf download

New York: Wiley. Chiang, S. Microstructure Development in Schaum's Outline of Mathematical Handbook of Formulas and tables , 4th ed. Spiegel, M. Hull, J. Schaum's Outline of Discrete Mathematics lets you focus on the problems that are at the heart of the subject.

Gradshteyn and I. Spiegel Murray R. Mathematical Handbook of Formulas and Tables. Schaum's Outline Series in Mathematics. Strang Gilbert Strang. Linear Algebra and Its Applications, third edition. Linear Algebra and Its Applications , third edition. Use Schaum's! Mathematical Handbook of Formulas and Tables , 2nd Ed. Mathematical Methods for Business and Economics This is called the principal branch and the values for this branch are called principal values.

Solid portions of curves correspond to principal values. Similar relations involving angles B and C can be obtained. See also formula 7. Sides a, b, c which are arcs of great circles are measured by their angles subtended at center O of the sphere.

A, B, C are the angles opposite sides a, b, c, respectively. Then the following results hold. Similar results hold for other sides and angles. Any one of the parts of this circle is called a middle part, the two neighboring parts are called adjacent parts, and the two remaining parts are called opposite parts.

The sine of any middle part equals the product of the tangents of the adjacent parts. The sine of any middle part equals the product of the cosines of the opposite parts. Laws of Logarithms The common logarithm of N is denoted by log10 N or briefly log N. For numerical values of common logarithms, see Table 1.

The natural logarithm of N is denoted by loge N or In N. For numerical values of natural logarithms see Table 7. For values of natural antilogarithms i. Here i is the imaginary unit [see page 10]. Similarly we define the other inverse hyperbolic functions. The inverse hyperbolic functions are multiple-valued and as in the case of inverse trigo- nometric functions [see page 49] we restrict ourselves to principal values for which they can be considered as single-valued.

The following list shows the principal values unless otherwise indicated of the inverse hyperbolic func- tions expressed in terms of logarithmic functions which are taken as real valued.

The process of taking a derivative is called differentiation. Thus, As examples we observe that Extensions to functions of more than two variables are exactly analogous. Note that dz is a function of four variables, namely x, y, dx, dy, and is linear in the variables dx and dy. Since the derivative of a du constant is zero, all indefinite integrals differ by an arbitrary constant. For the definition of a definite integral, see The process of finding an integral is called integration.

The following list gives some transformations and their effects. It is assumed in all cases that division by zero is excluded. General Formulas Involving Definite Integrals b b b b Rectangular formula: b Then there equation are 3 cases. Case 1. Linear, nonhomogeneous second There are 3 cases corresponding to those of entry Such quantities are called scalars.

Other quantities such as force, velocity, and momentum require for their specification a direction as well as magnitude. Such quantities are called vectors. A vector is represented by an arrow or directed line segment indicating direction. 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 y Cross or Vector Product Fundamental results follow: i j k 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.

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. Mode The mode is the value or values which occur most often.

Namely: 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. 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.

Hence, by Here M. Sample range: xn — x1. There are three quartiles: the first or lower quartile, denoted by Q1 or QL; the second quartile or median, denoted by Q2 or M; and the third or upper quartile, denoted by Q3 or QU.

Specifically: 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. 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.

Using the table in Fig. Three such types of curve are discussed as follows. Then log a and log b are obtained from transformed data points. 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. 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 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. This theorem can be genealized as follows: 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: 0. That is, events A and B are independent if the occurrence of one of them does not influence the occur- rence of the other. 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. 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 pair xi, f xi , usually given by a table, is called the probability distribution or probability mass function of X.

The function F x has the following properties: