“Elliptic Integral of the First Kind”的概念、定义、习题解题方法及翻译-数学辞典

单词

Elliptic Integral of the First Kind

释义

Elliptic Integral of the First Kind

Let the Modulus satisfy . (This may also be written in terms of theParameter or Modular Angle .) The incomplete elliptic integral of thefirst kind is then defined as

(1)

Let

			(2)
			(3)


			(4)

Let

			(5)
			(6)

so the integral can also be written as


			(7)
			(8)

where

is the complementary Modulus.

The integral

(9)

which arises in computing the period of a pendulum,

is also an elliptic integral of the firstkind. Use

			(10)
			(11)

to write




			(12)

(13)

Now let

(14)

so the angle

is transformed to

(15)

which ranges from 0 to

varies from 0 to

. Taking the differential gives

(16)

(17)

Plugging this in gives


			(18)

(19)

Making the slightly different substitution

, so

leads to an equivalent, but more complicatedexpression involving an incomplete elliptic function of the first kind,


			(20)

Therefore, we have proven the identity

(21)

The complete elliptic integral of the first kind, illustrated above as a function of , is defined by

			(22)
			(23)
			(24)

			(25)
			(26)
			(27)

where

(28)

is the Nome (for

is the Hypergeometric Function, and

is a LegendrePolynomial.

satisfies the Legendre Relation

(29)

where

and

are complete elliptic integrals of the first and Second Kinds, and

and

are the complementary integrals. The modulus

is often suppressed for conciseness, so that

and

are often simply written

and

, respectively.

The Derivative of is

(30)

(31)

(32)

(Whittaker and Watson 1990, pp. 499 and 521).

See also Amplitude, Characteristic (Elliptic Integral), Elliptic Integral Singular Value, Gauss'sTransformation, Landen's Transformation, Legendre Relation, Modular Angle, Modulus (EllipticIntegral), Parameter

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Elliptic Integrals.'' Ch. 17 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587-607, 1972.

Spanier, J. and Oldham, K. B. ``The Complete Elliptic Integrals and '' and ``The Incomplete Elliptic Integrals and .'' Chs. 61-62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 609-633, 1987.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

随便看

数学辞典收录了8975条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。