Elliptic functions
(gcide) | Function \Func"tion\, n. [L. functio, fr. fungi to perform,
execute, akin to Skr. bhuj to enjoy, have the use of: cf. F.
fonction. Cf. Defunct.]
1. The act of executing or performing any duty, office, or
calling; performance. "In the function of his public
calling." --Swift.
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2. (Physiol.) The appropriate action of any special organ or
part of an animal or vegetable organism; as, the function
of the heart or the limbs; the function of leaves, sap,
roots, etc.; life is the sum of the functions of the
various organs and parts of the body.
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3. The natural or assigned action of any power or faculty, as
of the soul, or of the intellect; the exertion of an
energy of some determinate kind.
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As the mind opens, and its functions spread. --Pope.
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4. The course of action which peculiarly pertains to any
public officer in church or state; the activity
appropriate to any business or profession.
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Tradesmen . . . going about their functions. --Shak.
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The malady which made him incapable of performing
his
regal functions. --Macaulay.
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5. (Math.) A quantity so connected with another quantity,
that if any alteration be made in the latter there will be
a consequent alteration in the former. Each quantity is
said to be a function of the other. Thus, the
circumference of a circle is a function of the diameter.
If x be a symbol to which different numerical values can
be assigned, such expressions as x^2, 3^x, Log. x, and
Sin. x, are all functions of x.
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6. (Eccl.) A religious ceremony, esp. one particularly
impressive and elaborate.
Every solemn `function' performed with the
requirements of the liturgy. --Card.
Wiseman.
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7. A public or social ceremony or gathering; a festivity or
entertainment, esp. one somewhat formal.
This function, which is our chief social event. --W.
D. Howells.
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Algebraic function, a quantity whose connection with the
variable is expressed by an equation that involves only
the algebraic operations of addition, subtraction,
multiplication, division, raising to a given power, and
extracting a given root; -- opposed to transcendental
function.
Arbitrary function. See under Arbitrary.
Calculus of functions. See under Calculus.
Carnot's function (Thermo-dynamics), a relation between the
amount of heat given off by a source of heat, and the work
which can be done by it. It is approximately equal to the
mechanical equivalent of the thermal unit divided by the
number expressing the temperature in degrees of the air
thermometer, reckoned from its zero of expansion.
Circular functions. See Inverse trigonometrical functions
(below). -- Continuous function, a quantity that has no
interruption in the continuity of its real values, as the
variable changes between any specified limits.
Discontinuous function. See under Discontinuous.
Elliptic functions, a large and important class of
functions, so called because one of the forms expresses
the relation of the arc of an ellipse to the straight
lines connected therewith.
Explicit function, a quantity directly expressed in terms
of the independently varying quantity; thus, in the
equations y = 6x^2, y = 10 -x^3, the quantity y is an
explicit function of x.
Implicit function, a quantity whose relation to the
variable is expressed indirectly by an equation; thus, y
in the equation x^2 + y^2 = 100 is an implicit
function of x.
Inverse trigonometrical functions, or Circular functions,
the lengths of arcs relative to the sines, tangents, etc.
Thus, AB is the arc whose sine is BD, and (if the length
of BD is x) is written sin ^-1x, and so of the other
lines. See Trigonometrical function (below). Other
transcendental functions are the exponential functions,
the elliptic functions, the gamma functions, the theta
functions, etc.
One-valued function, a quantity that has one, and only one,
value for each value of the variable. -- {Transcendental
functions}, a quantity whose connection with the variable
cannot be expressed by algebraic operations; thus, y in
the equation y = 10^x is a transcendental function of x.
See Algebraic function (above). -- {Trigonometrical
function}, a quantity whose relation to the variable is the
same as that of a certain straight line drawn in a circle
whose radius is unity, to the length of a corresponding
are of the circle. Let AB be an arc in a circle, whose
radius OA is unity let AC be a quadrant, and let OC, DB,
and AF be drawnpependicular to OA, and EB and CG parallel
to OA, and let OB be produced to G and F. E Then BD is the
sine of the arc AB; OD or EB is the cosine, AF is the
tangent, CG is the cotangent, OF is the secant OG is the
cosecant, AD is the versed sine, and CE is the coversed
sine of the are AB. If the length of AB be represented by
x (OA being unity) then the lengths of Functions. these
lines (OA being unity) are the trigonometrical functions
of x, and are written sin x, cos x, tan x (or tang x), cot
x, sec x, cosec x, versin x, coversin x. These quantities
are also considered as functions of the angle BOA.
Function |