# Math Cad 2000 Reference Manual

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M a t hca d 2 0 0 0Reference Manual

M a t h Soft , I n c. 1 0 1 M a in St r ee t Ca m bridge M a ssa chu se t t s 0 2 1 4 2 USA ht t p:/ / w w w . m a t hsoft . com /

MathSo f t + =

MathSoft, Inc. owns both the Mathcad software program and its documentation. Both the program and documentation are copyrighted with all rights reserved by MathSoft. No part of this publication may be produced, transmitted, transcribed, stored in a retrieval system, or translated into any language in any form without the written permission of MathSoft, Inc. U.S. Patent Numbers 5,526,475 and 5,468,538. See the License Agreement and Limited Warranty for complete information. English spelling software by Lernout & Hauspie Speech Products N.V. MKM developed by Waterloo Maple Software.

Copyright 1999 MathSoft, Inc. All rights reserved. MathSoft, Inc. 101 Main Street Cambridge, MA 02142 USA

Mathcad and Axum are registered trademarks and Electronic Book, QuickSheets, MathConnex, ConnexScript, Collaboratory, and the MathSoft logo are trademarks of MathSoft, Inc. Microsoft, Windows, and the Windows logo are registered trademarks of Microsoft Corp. Windows NT is a trademark of Microsoft Corp. MATLAB is a registered trademark of The MathWorks, Inc. Smart Sketch is a registered trademark of Intergraph Corporation. Other brand and product names referred to are trademarks or registered trademarks of their respective owners.

Printed in the United States of America. First printing: September 1999. Second Printing: November 1999.

Con t e nt sI n t r odu ct ion Notation Ch a p t e r 1 Fu n ct io n s Function Categories Finding More Information About the References Functions Ch a p t e r 2 Op e r a t or s Accessing Operators Finding More Information About the References Arithmetic Operators Matrix Operators Calculus Operators Evaluation Operators Boolean Operators Programming Operators (Professional) Ch a p t e r 3 Sym bolic Ke y w o r d s Accessing Symbolic Keywords Finding More Information Keywords App e n d ix AOt h e r Sp e cia l Fun ct ions Function Definitions Comments App e ndix BRe fe r e n ce s I nd e x 1 1 3 3 4 4 5 127 127 128 128 129 133 136 143 148 150 153 153 154 155 165 165 166 167 169

Content s

iii

IntroductionThis book lists and describes Mathcads built-in functions, operators, and symbolic keywords, emphasizing their mathematical and statistical aspects. Certain utility functions (for example, the various file access functions used for image processing) are described in greater depth in the Mathcad Users Guide.

NotationThroughout this book, the following notation is used whenever possible:

x and y represent real numbers. z and w represent either real or complex numbers. m, n, i, j, and k represent integers. S and any names beginning with S represent string expressions. u, v, and any names beginning with v represent vectors. A and B represent matrices or vectors. M and N represent square matrices. f represents a scalar-valued function. F represents a vector-valued function. file is a string variable that corresponds to a filename or path. X and Y represent variables or expressions of any type.

Notation

1

Cha pt e r 1 Fun ct ionsThis chapter lists and describes Mathcads built-in mathematical and statistical functions. The functions are listed alphabetically. Functions labeled Professional are available only in Mathcad Professional. Certain features described here accompany the Solving and Optimization Extension Pack (Expert Solver) , which requires Mathcad Professional and is available for sale separately. Function names are case-sensitive, but not font-sensitive. Type them in any font, but use the same capitalization as shown in the syntax section. Many functions described here as accepting scalar arguments will, in fact, accept vector arguments. For example , while the input z for the acos function is specified as a real or complex number, acos will in fact evaluate correctly at each of a vector input of real or complex numbers. Other functions may possess optional arguments, for example, cum int or fv . For such functions f and g , the notation f(x,[y]) means that y can be omitted, while the notation g(x,[[y],[z]]) means that both x and y can be omitted (but not just x or just y). Some functions dont accept input arguments with units. For such a function f, an error message must be dimensionless will arise when evaluating f(x), if x has units.

Fu nct ion Ca t e gor ie sEach function falls within one of the following categories:

Bessel Complex numbers Differential equation solving Expression type File access Finance Fourier transform Hyperbolic Interpolation and prediction Log and exponential Number theory/combinatorics Piecewise continuous Probability density Probability distribution Random number Regression and smoothing SolvingFunct ion Cat egories 3

Sorting Special Statistics String Trigonometric Truncation and round-off Vector and matrix Wavelet transform The category name is indicated in the upper right corner of each entry. To see all the functions that belong to a given category, check the index of this book.

Fin din g M or e I nform a t ionYou can also find information about functions using either of these methods:

To quickly see a short description of each function from within Mathcad, choose Function from the Insert menu. Select a function in the Function field, then read the description in the Description field. Click on the Help button to see the Help topic on a selected function. Refer to the Resource Center QuickSheets for more detailed information about functions, categories, and related topics. Select Resource Center from the Help menu. Then click on the QuickSheets icon and select a specific topic.

About t h e Re fe r e n ce sReferences are provided in Appendix B for you to learn more about the numerical algorithm underlying a given Mathcad function or operator. References are not intended to give a description of the actual underlying source code. Some references (such as Numerical Recipes) do contain actual C code for the algorithms discussed therein, but the use of the reference does not necessarily imply that the code is what is implemented in Mathcad. The references are cited for background information only.

4

Chapt er 1 Funct ions

Fu nct ion sa co sSy nt ax Descript ion Argum ent s zacos( z)

Trigonom et ric Returns the inverse cosine of z (in radians). The result is between 0 and if z is real. For complex z, the result is the principal value. real or complex number Hyperbolicacosh( z)

a co shSy nt ax Descript ion Argum ent s z

Returns the inverse hyperbolic cosine of z. The result is the principal value for complex z. real or complex number Trigonom et ricacot ( z)

a co tSy nt ax Descript ion Argum ent s z

Returns the inverse cotangent of z (in radians). The result is between 0 and if z is real. For complex z, the result is the principal value. real or complex number Hyperbolicacot h( z)

a co t hSy nt ax Descript ion Argum ent s z

Returns the inverse hyperbolic cotangent of z. The result is the principal value for complex z. real or complex number Trigonom et ricacsc( z)

a cscSy nt ax Descript ion Argum ent s z

Returns the inverse cosecant of z (in radians). The result is the principal value for complex z. real or complex number Hyperbolicacsch( z)

a cschSy nt ax Descript ion Argum ent s z

Returns the inverse hyperbolic cosecant of z. The result is the principal value for complex z. real or complex number

Functions

5

AiSy nt ax Descript ion Argum ent s x Exam ple

( Professional)Ai( x)

Bessel

Returns the value of the Airy function of the first kind. real number

Com m ent s Algorit hm See also

This function is a solution of the differential equation:

d yxy = 0. d x2 Asymptotic expansion (Abramowitz and Stegun, 1972)

2

Bi

a n g leSy nt ax Descript ion Argum ent s x, y See alsoangle( x, y)

Trigonom et ric Returns the angle (in radians) from positive x-axis to point (x, y) in x-y plane. The result is between 0 and 2. real numbersarg , at an , at an2

APPEN D PRNSy nt ax Descript ionAPPENDPRN( file) : = A

File Access

Appends a matrix A to an existing structured ASCII data file. Each row in the matrix becomes a new line in the data file. Existing data must have as many columns as A . The function must appear alone on the left side of a definition. string variable corresponding to structured ASCII data filename or pathWRI TEPRN for more details

Argum ent s file See also

6

Chapt er 1 Funct ions

argSy nt ax Descript ion Argum ent s z See alsoarg( z)

Com plex Num bers Returns the angle (in radians) from the positive real axis to point z in the complex plane. The result is between and . Returns the same value as that of when z is written as r e i . real or complex numberangle , at an , at an2

a se cSy nt ax Descript ion Argum ent s zasec( z)

Trigonom et ric Returns the inverse secant of z (in radians). The result is the principal value for complex z. real or complex number Hyperbolicasech ( z)

a se chSy nt ax Descript ion Argum ent s z

Returns the inverse hyperbolic secant of z. The result is the principal value for complex z. real or complex number Trigonom et ricasin ( z)

a sinSy nt ax Descript ion Argum ent s z

Returns the inverse sine of z (in radians). The result is between /2 and /2 if z is real. For complex z, the result is the principal value. real or complex number Hyperbolicasinh ( z)

a sin hSy nt ax Descript ion Argum ent s z

Returns the inverse hyperbolic