QMTools recognizes several mathematical and physical constants, as well as numerous operators and functions. The latter include most commonly used functions of mathematics, as well as a variety of special functions important in scientific applications.

Reserved Constants

The mathematical constants e (base of natural logarithms) and π (Euclidean ratio of circle circumference to diameter) have predefined values.

• e = 2.718281828459045...
• π = 3.141592653589793...

The value of these constants can always be viewed by entering the appropriate symbol on the Math Palette (type alt+P to insert π), followed by ctrl+Z.

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Standard Mathematical Functions

Most functions accept a complex argument, indicated by "z" in the usage column below; all others take real arguments only.

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Special Function Support

Special function values are computed to an accuracy of better than 1 part in 10⁶ (and often much better than this).

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Operators

Operator support extends beyond the familiar arithmetic operations (addition, multiplication, etc.), to include roots, conditionals, and summation.

• Arithmetic operations are inserted in context by typing the expected characters: '+' (addition), '−' (subtraction), '*' (multiplication), '/' (division), and '^' (exponentiation).
  Multiple operations are evaluated in the order of their precedence, which can be clarified or overridden by surrounding parentheses. An existing node may be enclosed in parentheses by selecting it and typing ctrl+9.
• Roots may be found using the operator for exponentiation: for example, you would enter 2.5^0.25 to calculate the 4th root of 2.5. For the special case of square roots, QMTools offers the added convenience of the root symbol (available from the Insert item of the formula popup menu) and filling in the operand.
• Conditionals refer to structures where the result is contingent upon a condition.

  Example:

  The function f(x) which is zero to the left of the origin and unity to the right (Heaviside step function) is defined on the Math Palette by the following statement:

  f(x) ≡ x < 0 → 0 | 1

  This statement has the general form:

  expr1 → expr2 | expr3

  where expr1 refers to a boolean expression. The right arrow is made by typing ctrl+A; typing the backslash character '\' inserts the vertical bar. Evaluation proceeds as follows: expr1 is evaluated; if the value is true, then the value of expr2 is returned; otherwise the value of expr3 is returned.

  Conditional statements also can be nested to produce more complicated forms, as illustrated below.

  Example:

  A function g(x) that evaluates to unity for 0 < x < 1 and is zero otherwise would be defined on the Math Palette by the statement:

  g(x) ≡ x ≤ 0 → 0 | x < 1 → 1 | 0
• Summation requires a special 'under-over' structure that is readily inserted by typing the keyboard shortcut alt+ctrl+S.

  Example:

  To generate the formula for summing the quantity 2n + 1 from n = 0 to n = 4, you would do the following:

  1. Type alt+ctrl+S to insert the summation symbol, along with a template for the limit fields and the expression to be summed (the summand). The cursor is left in the limit stop field.
  2. Type "4", followed by a right arrow to move the cursor to the limit start field.
  3. Next type "0", followed by the up arrow to move the cursor to the summand field.
  4. Lastly, type the open parenthesis "(" (creates the group consisting of both an open and a close), followed by the summand "2n + 1".

  Once the full summation structure is entered, the sum is evaluated by typing ctrl+Z.
The summation index is identified with the independent [unassigned] variable of the summand, and is incremented from the lower to the upper limit in steps equal to the grid size that has been established for this variable's domain. In particular, the increment is not unity unless it has been so specified (see Variables for how to edit a variable's domain).

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