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Parameter definition (http://mathinsight.org/definition/parameter)

A parameter is a quantity that influences the output or behavior of a mathematical

object but is viewed as being held constant. Parameters are closely related to variables, and the

difference is sometimes just a matter of perspective. Variables are viewed as changing while

parameters typically either don't change or change more slowly. In some contexts, one canimagine performing multiple experiments, where the variables are changing through each

experiment, but the parameters are held fixed during each experiment and only change

between experiments.

One place parameters appear is within functions. For example, a function might a

generic quadratic function as

f(x)=ax2+bx+c.f(x)=ax2+bx+c.

Here, the variable xx is regarded as the input to the function. The symbols aa, bb,and cc are parameters that determine the behavior of the function ff. For each value of the

parameters, we get a different function. The influence of parameters on a function is captureed

by the metaphor of dials on a function machine. 

Parameter (http://mathworld.wolfram.com/Parameter.html) 

The term "parameter" is used in a number of ways in mathematics. In general, mathematical

functions may have a number of arguments. Arguments that are typically varied when plotting,

performing mathematical operations, etc., are termed "variables," while those that are not

explicitly varied in situations of interest are termed "parameters." For example, in the standard

equation of an ellipse 

(1)

and are generally considered variables and and are considered parameters. Thedecision on which arguments to consider variables and which to consider parameters may be

historical or may be based on the application under consideration. However, the nature of a

mathematical function may change depending on which choice is made. For example, the above

equation is quadratic in and , but if and are instead considered as variables, the resulting

equation

(2)

is quartic in and .

http://mathinsight.org/definition/variablehttp://mathinsight.org/definition/variablehttp://mathinsight.org/definition/variablehttp://mathinsight.org/function_machine_parametershttp://mathinsight.org/function_machine_parametershttp://mathinsight.org/function_machine_parametershttp://mathworld.wolfram.com/Variable.htmlhttp://mathworld.wolfram.com/Variable.htmlhttp://mathworld.wolfram.com/Ellipse.htmlhttp://mathworld.wolfram.com/Ellipse.htmlhttp://mathworld.wolfram.com/Ellipse.htmlhttp://mathworld.wolfram.com/Ellipse.htmlhttp://mathworld.wolfram.com/Variable.htmlhttp://mathinsight.org/function_machine_parametershttp://mathinsight.org/definition/variable

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In the theory of elliptic integrals, "the" parameter is denoted and is defined to be

(3)

where is the elliptic modulus.  An elliptic integral is written when the parameter is

used, whereas it is usually written where the elliptic modulus is used. The elliptic

modulus tends to be more commonly used than the parameter (Abramowitz and Stegun 1972,

p. 337; Whittaker and Watson 1990, p. 479), although most of Abramowitz and Stegun (1972,

pp. 587-607), i.e., the entire chapter on elliptic integrals, and the Wolfram

Language's EllipticE, EllipticF, EllipticK, EllipticPi, etc., use the parameter.

The complementary parameter is defined by

(4)

where is the parameter.

Let be the nome,  the elliptic modulus, where . Then

(5)

where is the complete elliptic integral of the first kind,  and . Then theinverse of is given by

(6)

where is a Jacobi theta function. 

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