Reference

It is recommended to always use the iMath menus to create or modify iFormulas. Nevertheless, sometimes it will be necessary to fine-tune equations by editing them directly.

All iMath commands start with the string %%ii at the beginning of a line, which will be interpreted as a comment by Openoffice. When the document is recalculated, iMath parses the string and generates the formula that is displayed. The generated lines are marked by %%gg at the end of the line. Editing such a line makes no sense because it will be deleted at the next recalculation! Always modify the %%ii lines if you want to tweak an iFormula. Formulas may also have lines without either %%ii or %%gg, these will not be touched by iMath.

There is also a reference section integrated into the Office help. Press F1 and enter "iFormula" into the Index tab of the help.

Elements

The following elements can be part of an iFormula definition string

• symbol

The name of a mathematical symbol or function, e.g. x or a_n

• unit

The name of a physical unit, either preceded by a percent sign or enclosed in quotation marks, e.g. %meter or "meter"

• expression

A mathematical expression in Openoffice math format

• equation

A kind of expression which has the form expression1 = expression2

• vector = expression1 : expression2 : expression3

A vector of expressions where expression1 is the first element, expression2 the second element and the optional expression3 the step. By default the given interval will be subdivided into 20 steps, so for example x = 1:20 will give a vector containing the integer numbers from 1 to 20.

• symbol [ number ] or symbol [ number, number ]

Accesses an element of a vector or matrix.

• @label@

A equation label, used to reference a previously defined equation. The label MUST be enclosed in hash signs.

• list = { element; element; ... }

A list of other elements, enclosed in parentheses and separated by semicolons

• *

If the asterisk is given, then the output of this line is supressed, which is useful for hiding intermediate calculations.

• { option; option; ... }

The options can be any of those described for the OPTIONS keyword. It is not necessary to specify options (you need to leave away the brackets, too, if you don't).

Commands

After the initial %%ii, several groups of commands are possible.

Definitions

iMath allows you to define equations, units and functions, and to evaluate expressions.

{ options } EXDEF expression

Create an expression. This keyword is most frequently used with VAL to find the values of a symbol or an expression, e.g. EXDEF VAL(y).

@label@ { options } EQDEF * expression

Define an equation. The expression most be of the form expression1 = expression2. For the special case of the equation label starting with lib:, the equation will not be erased by CLEAREQUATIONS. This is useful for building a library of generic equations, as for example in the file substitutions.imath.

@label@ { options } CONSTDEF * expression

Define a constant. The expression most be of the form variable = expression, and the expression must evaluate to a quantity (a numeric with or without units). Constants are not erased by CLEAREQUATIONS. This makes it possible to specify general constants only once in the file and reuse them for several different calculations.

@label@ { options } FUNCDEF * funcname(args) = expression

Define an equation that can be used to find the value of a user-defined function. The function must first have been declared with FUNCTION.

FUNCTION { hints, symbol, expression }

Declare a user-defined function. symbol is the name of the function. The expression declares the arguments of the function and can be either a single symbol or a list of symbols. The way that iMath treats this function can be influenced by function hints:

• none: No function hints (must be used because iMath requires three arguments to the FUNCTION declaration
• lib: A library function: It will not be affected by CLEAREQUATIONS
• trig: For trigonometric functions. Tells iMath that this function will get special treatment when printing (e.g. <math>sin^2 x</math> instead of <math>(sin x)^2</math>)
• expand: The function is to be expanded immediately. Useful for functions that are really abbreviations, e.g. square(x) will be expanded to <math>x^2</math>
• nobracket: The function does not require brackets around the argument (e.g. sqrt(x) is printed as <math>root x</math>)
• defdiff: Differentiate the definition of the function, not the function itself

UNITDEF { "unitname", symbol = expression }

Defines a new unit. The symbol is the name of the unit, this must begin with a percent sign % to distinguish it from a variable name. The expression defines the unit in terms of units already known to iMath. If this expression contains data types apart from units, powers of units, numerics, or constants already known to iMath, a warning is issued, but the unit definition is created anyway. The unitname is optional and gives a string to be printed instead of the symbol. This is useful when defining units whose name is not a valid symbol, e.g. inches: UNITDEF{"''", %inch = 0.0254 %m}.

VECTORDEF symbol

Defines the symbol to be a vector. This is important because vectors are non-commutative when evaluating an expression.

@label@ { options } VECTORDEF * symbol = expression

Defines the symbol to be a vector and assigns the expression to it. It is important to use this keyword because vectors are non-commutative when evaluating an expression.

MATRIXDEF symbol

Defines the symbol to be a (two-dimensional) matrix. This is important because matrices are non-commutative when evaluating an expression.

@label@ { options } MATRIXDEF * symbol = expression

Defines the symbol to be a (two-dimensional) matrix and assigns the expression to it. It is important to use this keyword because matrices are non-commutative when evaluating an expression.

TEXT string

Arbitrary text that should be displayed between two iMath formulas. The text must be enclosed in quotes, and formatting rules for Office formulas apply.

DELETE { @label1@; @label2@; ... }

Removes an equation such that it has no influence on future searches for variable values. Note that the equation is not deleted from the file, but only becomes ineffective (and unaccessible) after the point in the text where the delete statement is.

CLEAREQUATIONS

iMath stores all equations and functions that it encounters. If you want to make a fresh start in your document, use this command.

Symbolic manipulations of expressions

DIFFERENTIAL ( expression )

Create a differential object. Please note that though you can handle this object like a variable, this does not always make mathematical sense!

DIFFERENTIATE ( equation|expression, variable, variable)

Differentiate the expression or both sides of the equation with respect to the variable. The second variable gives the number of times to differentiate.

INTEGRATE ( equation|expression, variable1, variable2)

Create an integral for the expression or for both sides of the equation with respect to variable1. variable2 is the integration constant. The integral will be evaluated immediately (if possible).

INTEGRATE ( equation, {variable;variable}, {variable;variable})

Create an integral for both sides of the equation with respect to the two variables in the first list. The second list contains the integration constants to be used. The integrals will be evaluated immediately (if possible).

INTEGRATE ( equation|expression, variable1, variable2, variable3)

Create an integral for the expression or for both sides of the equation with respect to variable1. Variable2 is the lower bound, variable3 is the upper bound. The integral will be evaluated immediately (if possible).

INTEGRATE ( equation, {variable;variable}, {variable;variable}, {variable;variable})

Create an integral for both sides of the equation with respect to the variables in the first list. The second list contains the lower bounds, the third contains the upper bounds. The integrals will be evaluated immediately (if possible).

INT FROM variable = expression TO { expresssion} { expression differential(variable)}

Create an integral from the lower bound to the upper bound. The function can then be symbolically evaluated with the SIMPLIFY keyword. If the upper bound is a number or a variable then the curly braces around it can be omitted. Example: INT FROM x = -%pi to %pi { sin(x) differential(x)}. The differential can be contained in the expression and iMath will find it (you might have to expand the expression first). But if there is no differential matching the integration variable anywhere then you will get strange results.

INT { expression differential(variable)}

Create an integral without bounds. The expression must be enclosed in brackets. The function can then be symbolically evaluated with the SIMPLIFY keyword.

ITERVAL ( expression, expression, expression, expression, integer )

Iterate an expression until it converges. Example: ITERVAL(x, sqrt(x), 100, 0.1, 10) will start iterating with x=100. The next value of x will be x=sqrt(100). Iteration will continue until the difference between two consecutive values of x is less than 0.1. The last parameter gives the maximum number of iteration, in case there is no convergence.

ITERVAL ( list, list, list, list, integer )

Iterate a list of expressions until it converges. Example: ITERVAL(x;y, sqrt(x);ln(y), 100;20, 0.1;0.01, 10) will start iterating with x=100 and y=20. The next value of x and y will be x=sqrt(100) and y=ln(20). Iteration will continue until the difference between two consecutive values of x and y is less than 0.1 and 0.01, respectively. The last parameter gives the maximum number of iteration, in case there is no convergence.

LHS ( equation )

Gets the left hand side of the equation.

NROOT number number

Creates the nth root of a number. If you specify expressions instead of numbers, you must enclose them in curly brackets {} to avoid ambiguities.

NUMVAL ( expression )

Try to find a numeric value (without units) for the expression from any equations defined previously in the document. Gives an error if the expression does not evaluate to a number.

NUMVALWITH ( expression, expression; expression; ... )

Try to find a numeric value for the expression from any equations defined previously in the document, plus the equations given as the second argument. Gives an error if the expression does not evaluate to a number.

PDIFFERENTIATE ( equation|expression, variable, variable)

Partially differentiate the expression or both sides of the equation with respect to the variable. The second variable gives the number of times to differentiate.

QUANTITY ( expression )

Try to find a quantity (number with optional units) for the expression from any equations defined previously in the document. Gives an error if the expression does not evaluate to a quantity.

QUANTITYWITH ( expression, expression; expression; ... )

Try to find a quantity (number with optional units) for the expression from any equations defined previously in the document, plus the equations given as the second argument. Gives an error if the expression does not evaluate to a quantity.

QUO ( expression, expression, variable )

Calculates the quotient of two polynomials in the given variable. An error message will be shown if one of the expressions is not a polynomial in the variable.

REM ( expression, expression, variable )

Calculates the remainder after division of two polynomials in the given variable. An error message will be shown if one of the expressions is not a polynomial in the variable.

REV ( equation )

Reverse the left hand side and the right hand side of the equation

RHS ( equation )

Gets the right hand side of the equation

SIMPLIFY ( equation|expression, { "simplification"; "simplification"; ... } )

Simplifies both sides of the equation according to the simplifications given in a list. Possible values are:

• expand Fully expands all expressions, including function arguments.
• expandf Only expand function definition, not arguments.
• eval Numerically evaluate the equation as far as possible.
• normal Normalize the equation (see description of GiNaC normal() method for details).
• collect-common Collect common factors (see description of GiNaC collect_common_factors() method for details).
• unsafe Does unsafe simplifications, for example sqrt{x^2} => x or \arctan\tan{x} => x. Note that the opposite \tan\arctan{x} => x is not an unsafe evaluation and is thus done automatically (by GiNaC).
• diff Evaluates differential function diff, thus symbolically calculating differentials as far as possible.
• sum Evaluates sum functions, thus symbolically calculating sums as far as possible.
• gather-sqrt Gathers single square roots under one root symbol.
• integrate Symbolically integrates any integrals in the expression as far as possible.

COLLECT ( equation|expression, variable )

Tries to restructure both sides of the equation as a polynomial in the given variable. The variable may be omitted, in this case, iMath uses the variable that occurs in the greatest amount of powers.

SOLVE ( equation, variable, solution )

Solve the equation for the variable. If there are several possible solutions (e.g. for a quadratic equation), you can specify which one you want in the third argument. Note that this argument is not optional, so you must specify 1 for a linear equation.

SUBST ( expression, equation OR {equation; equation; ...} )

Substitute the equation or the list of equations in the expression. All substitutions will be done in parallel, which means that a list of equations like { x = y; y = z} might not have the desired effect (consider using SUBSTC instead).

SUBSTC ( expression, equation OR {equation; equation; ...} )

Substitute the equation or the list of equations in the expression. The substitutions will be done in the order you specifiy them, so a list of equations like { x = y; y = z} will have the effect of substituting x=z.

SUBSTV ( expression, variable = vector )

Substitute a vector of expressions in the expression. The result will be a vector. This is mostly useful for specifying values to the CHART keyword.

SUM FROM variable = expression TO { expresssion} { expression }

Create a sum function summing up the expression from the lower bound to the upper bound. The function can then be symbolically evaluated with the SIMPLIFY keyword. If the upper bound is a number or a variable then the curly braces around it can be omitted.

TSERIES ( expression, variable = expression, nth )

Create a Taylor series of nth grade for the expression around the given point.

UNIT ( expression )

Try to find a quantity (number with optional units) for the expression from any equations defined previously in the document, and then get the units of this result. Gives an error if the expression does not evaluate to a quantity.

UNITWITH ( expression, expression; expression; ... )

Try to find a quantity (number with optional units) for the expression from any equations defined previously in the document, plus the equations given as the second argument, and then get the units of this result. Gives an error if the expression does not evaluate to a quantity.

VAL ( expression )

Try to find a numeric value for the expression from any equations defined previously in the document.

VALWITH ( expression, expression; expression; ... )

Try to find a numeric value for the expression from any equations defined previously in the document, plus the equations given as the second argument. These will be removed again after the value search, so they have no effect on the rest of the document.

Charts

CHART { "objectname", xval, xunit, yval, yunit, idx }

Update or create data in a chart with the Office object name given in the string (in other words, the chart must already exist in the document). The x-values can be either a vector of values or an equation assigning a vector of values to a variable. The x-values will be expressed in the given units. In other words, after dividing the x-values by the units, a vector of purely numeric values must result. The y-values can be either a vector of values or an equation or an equation label. If you specifiy an equation or a label, the vector of x-values will be substituted into the equation to produce a vector of y-values. This will only work if you specified the x-values as an equation! If the y-values have units, they must be specified. The last argument is the index of the chart series. This enables you to add or manipulate different series of the chart.

Reading and writing files

READFILE { "path/to/file" }

Open the given file and parse the iMath commands in this file. This command can be used to put common formulas in a file to re-use them. At the moment, the path is operation-system dependent! In other words, you cannot use an Office URL.

Options

Options can be defined with the keyword OPTIONS { key1 = value1; key2 = value2; ... }. The following key-value pairs are possible.

Default units

units={unit1; unit2; ...} Specifies a list of units separated by ';' which should be used for printing physical quantities. If no such list is specified, the SI base units are used. A useful list of units for mechanical engineering might be {%mm; %N; %Nm; %Nosqmm}. The order in which the units are specified is important! Simple units should be specified first, more complex ones later on. The reason is that the complex units usually imply the simpler ones, for example, N implies mm because 1 N equals 1 kg m/s^2.

Suppress display of units

suppress_units=true|false Specifies whether units should be suppressed when printing formulas or evaluation resaults. The default is suppress_units=false. Note that calculations will always be made with units. This option simply suppresses unit display.

Precision of numbers

precision=integer Specifies what precision is to be used for printing floating point numbers. The precision is the number of digits printed, more digits are rounded away. The precisions is 4 by default. The next option fixedpoint influences the interpretation of the precision.

Fixed or floating point notation of numbers

fixedpoint=true/false Specifies whether floating point numbers should be printed with a fixed number of digits after the decimal marker (true) or with a fixed number of significant digits, not counting leading zeros (false). The default is fixedpoint=false.

Fixed exponent

exponent=integer Specifies that all numbers should be printed with the given exponent

Display of differentials

difftype="line"/"dot"/"dfdt" Specifies how differentials should be printed: With an apostrophe after the variable name (f'), with a dot on top of the variable name (dot f) or as df/dt.

Automatic formatting of iFormulas

autoformat=true/false Specifies whether iMath should do automatic formatting of your input equations. By default this option is turned off (false, meaning that the equations are printed exactly as you typed them in). This of course is only possible for original equations, not for equations obtained by symbolic operations.

Automatic aligning of iFormulas

autoalign=true/false Specifies whether iMath should automatically align multi-line formulas at the equation symbol. Consecutive formulas must be merged (e.g. using the context menu) for this option to have any effect.

Automatic chaining of iFormulas

autochain=true/false Specifies whether iMath should omit the left hand side of an equation in multi-line formulas if it is identical to the previous line's left hand side.

Inhibit underflow exceptions

underflow=true|false Specifies whether iMath should silently ignore underflow exceptions. In this case numbers smaller than the underflow limit will be treated as zero.

Evaluate odd nth roots of negative bases to the real value

realroots=true|false Specifies whether iMath should return real nth roots as the result of VAL() where the base is negative. By default, the root on the principal branch is returned (which is complex for negative bases). For example, the third root of -1 would be returned as 1/2 + 1/2 * sqrt(3) * I.

Miscellaneous

SIZE sizestring

Change the default display size of the formula (the usage is identical with normal math formula usage).

ALIGNL

Align the following text to the left (use bracketing if you need to align more than one symbol).

ALIGNC

Align the following text to the center.

ALIGNR

Align the following text to the right.

Built-in constants and functions

This section describes constants and functions specific to iMath. Most functions will try to immediately evaluate themselves. If this is not possible, then the function will remain in the expression where it has been used. In that case it can be evaluated later with the SIMPLIFY keyword

Constants

iMath knows about Pythagoras' number %pi and the basis of the natural logarithm %e. Trigonometric expressions with pi or multiples of it will usually simplify automatically, but the expression ln(%e) requires a VAL() call to evaluate to 1.

Functions for vectors and matrices

vmin ( vector )

Immediately evaluates to the minimal element of the vector if possible.

vmax ( vector )

Immediately evaluates to the maximal element of the vector if possible.

transpose ( vector or matrix )

Immediately evaluates to the transposed vector or matrix.

Functions for number display

round ( expression; expression )

Immediately evaluates to a number rounded to the given number of decimal places, if possible.

ceil ( expression; expression )

Immediately evaluates to a number rounded up to the given number of decimal places, if possible.

floor ( expression; expression )

Immediately evaluates to a number rounded down to the given number of decimal places, if possible.

Functions for conditional evaluation

ifelse ( condition; expression1; expression2 )

Immediately evaluate to expression1 or expression2 if the condition is an equation involving numbers (e.g. 3 < 4).