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

单词

Elliptic Integral

释义

Elliptic Integral

An elliptic integral is an Integral of the form

(1)

(2)

where

, and

are Polynomials in

and

is a Polynomial of degree 3 or 4. Another form is

(3)

where

is a Rational Function of

and

is a function of

Cubic orQuadratic in

contains at least one Odd Power of

, and

has norepeated factors.

Elliptic integrals can be viewed as generalizations of the inverse Trigonometric Functions and providesolutions to a wider class of problems. For instance, while the Arc Length of a Circle is given as a simplefunction of the parameter, computing the Arc Length of an Ellipse requires an elliptic integral. Similarly,the position of a pendulum is given by a Trigonometric Function as afunction of time for small angle oscillations, but the full solution for arbitrarily large displacements requires the useof elliptic integrals. Many other problems in electromagnetism and gravitation are solved by elliptic integrals.

A very useful class of functions known as Elliptic Functions is obtained by invertingelliptic integrals to obtain generalizations of the trigonometric functions. Elliptic Functions(among which the Jacobi Elliptic Functions and Weierstraß Elliptic Function are the two most common forms) provide a powerful tool for analyzing many deep problems in Number Theory,as well as other areas of mathematics.

All elliptic integrals can be written in terms of three ``standard'' types. To see this, write

(4)

But since

(5)

then


	(6)

(7)

But any function

can be evaluated in terms of elementary functions, so the only portion that need beconsidered is

(8)

Now, any quartic can be expressed as

where

			(9)
			(10)

The Coefficients here are real, since pairs of Complex Roots are Complex Conjugates


			(11)

If all four Roots are real, they must be arranged so as not to interleave (Whittaker and Watson 1990, p. 514). Now define aquantity

such that

(12)

is a Square Number and

(13)

(14)

Call the Roots of this equation

and

, then



			(15)


			(16)

Taking (15)-(16) and

gives


			(17)

			(18)

Solving gives


			(19)

			(20)

so we have

(21)

Now let

			(22)


			(23)


			(24)

and

			(25)

			(26)

Now let

(27)

(28)

Rewriting the Even and Odd parts

			(29)
			(30)

gives

(31)

so we have

(32)

Letting

			(33)
			(34)

reduces the second integral to

(35)

which can be evaluated using elementary functions. The first integral can then be reduced by Integration by Parts toone of the three Legendre elliptic integrals (also called Legendre-Jacobi Elliptic Integrals),known as incomplete elliptic integrals of the first, second, and third kind, denoted

, and

, respectively (von Kármán and Biot 1940, Whittaker and Watson 1990, p. 515). If

, then the integralsare called complete elliptic integrals and are denoted

Incomplete elliptic integrals are denoted using a Modulus , Parameter, or Modular Angle . An elliptic integral is written when theParameter is used, when the Modulus is used, and when the Modular Angle is used. Complete elliptic integrals are defined when andcan be expressed using the expansion

(36)

An elliptic integral in standard form

(37)

where

(38)

can be computed analytically (Whittaker and Watson 1990, p. 453) in termsof the Weierstraß Elliptic Function withinvariants

			(39)
			(40)

is a root of

, then the solution is

(41)

For an arbitrary lower bound,



	(42)

where

is a Weierstraß Elliptic Function.

A generalized elliptic integral can be defined by the function

			(43)
			(44)

(Borwein and Borwein 1987). Now let

			(45)
			(46)

But

(47)



			(48)

and

(49)

and the equation becomes


			(50)

Now we make the further substitution

. The differential becomes

(51)

but

, so

(52)

(53)

and

(54)

However, the left side is always positive, so

(55)

and the differential is

(56)

We need to take some care with the limits of integration. Write (50) as

(57)

Now change the limits to those appropriate for the

integration

(58)

so we have picked up a factor of 2 which must be included. Using this fact and plugging (56) in (50) thereforegives

(59)

Now note that

			(60)
			(61)
			(62)

Plug (62) into (59) to obtain


			(63)

But

	(64)
	(65)
	(66)

(67)

and (63) becomes


			(68)

We have therefore demonstrated that

(69)

We can thus iterate

			(70)
			(71)

as many times as we wish, without changing the value of the integral. But this iteration is the same as and thereforeconverges to the Arithmetic-Geometric Mean, so the iteration terminates at

, and we have





			(72)

Complete elliptic integrals arise in finding the arc length of an Ellipse and the period of apendulum. They also arise in a natural way from the theory of Theta Functions. Complete elliptic integrals can be computed using a procedure involving the Arithmetic-Geometric Mean. Notethat



			(73)

So we have

(74)

where

is the complete Elliptic Integral of the First Kind. We are free to let

and

, so

(75)

since

, so

(76)

But the Arithmetic-Geometric Mean is defined by

			(77)
			(78)
			(79)

where

(80)

so we have

(81)

where

is the value to which

converges. Similarly, taking instead

and

gives

(82)

Borwein and Borwein (1987) also show that defining

(83)

leads to

(84)

(85)

for

and

, and

(86)

The elliptic integrals satisfy a large number of identities. The complementary functions and moduli are defined by

(87)

Use the identity of generalized elliptic integrals

(88)

to write



			(89)

(90)

Define

(91)

and use

(92)

(93)

Now letting

gives

(94)

(95)


			(96)

and


			(97)

Writing

instead of

(98)

Similarly, from Borwein and Borwein (1987),

(99)

(100)

Expressions in terms of the complementary function can be derived from interchanging the moduli and their complements in(93), (98), (99), and (100).



			(101)

(102)

and

(103)

(104)

Taking the ratios

(105)

gives the Modular Equation of degree 2. It is also true that

(106)

See also Abelian Integral, Amplitude, Argument (Elliptic Integral), Characteristic (EllipticIntegral), Delta Amplitude, Elliptic Function, Elliptic Integral of the First Kind, EllipticIntegral of the Second Kind, Elliptic Integral of the Third Kind, Elliptic Integral Singular Value, HeumanLambda Function, Jacobi Zeta Function, Modular Angle, Modulus (Elliptic Integral), Nome,Parameter

References

Elliptic Integrals

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.

Arfken, G. ``Elliptic Integrals.'' §5.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 321-327, 1985.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.

Hancock, H. Elliptic Integrals. New York: Wiley, 1917.

King, L. V. The Direct Numerical Calculation of Elliptic Functions and Integrals. London: Cambridge University Press, 1924.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Elliptic Integrals and Jacobi Elliptic Functions.'' §6.11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 254-263, 1992.

Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 1: Elementary Functions. New York: Gordon & Breach, 1986.

Timofeev, A. F. Integration of Functions. Moscow and Leningrad: GTTI, 1948.

von Kármán, T. and Biot, M. A. Mathematical Methods in Engineering: An Introduction to the Mathematical Treatment of Engineering Problems. New York: McGraw-Hill, p. 121, 1940.

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

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