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

单词

Inverse Tangent

释义

Inverse Tangent

The inverse tangent is also called the arctangent and is denoted either or arctan . It has theMaclaurin Series

(1)

A more rapidly converging form due to Euler

is given by

(2)

(Castellanos 1988). The inverse tangent satisfies

(3)

for Positive and Negative

, and

(4)

for

. The inverse tangent is given in terms of other inverse trigonometric functions by

			(5)
			(6)
			(7)

for Positive or Negative

, and

			(8)
			(9)
			(10)
			(11)

for

In terms of the Hypergeometric Function,

			(12)
			(13)

(Castellanos 1988). Castellanos (1986, 1988) also gives some curious formulas in terms of the Fibonacci Numbers,

			(14)
			(15)
			(16)

where

			(17)
			(18)

and

is the largest Positive Root of

(19)

The inverse tangent satisfies the addition Formula

(20)

as well as the more complicated Formulas

(21)

(22)

(23)

the latter of which was known to Euler.

The inverse tangent Formulas areconnected with many interesting approximations to Pi

(24)

Euler

gave

(25)

where

(26)

The inverse tangent has Continued Fraction representations

(27)

To find

numerically, the following Arithmetic-Geometric Mean-like Algorithm canbe used. Let

			(28)
			(29)

Then compute

			(30)
			(31)

and the inverse tangent is given by

(32)

(Acton 1990).

An inverse tangent with integral is called reducible if it is expressible as a finite sum of the form

(33)

where

are Positive or Negative Integers and

are Integers

is reducible Iff all the Prime factors of

occur among the Prime factors of

for

, ...,

. A second Necessary and Sufficient condition is that the largest Prime factorof

is less than

. Equivalent to the second condition is the statement that every Gregory Number

can be uniquely expressed as a sum in terms of

s for which

is a StérmerNumber (Conway and Guy 1996). To find this decomposition, write

(34)

so the ratio

(35)

is a Rational Number. Equation (35) can also be written

(36)

Writing (33) in the form

(37)

allows a direct conversion to a corresponding Inverse Cotangent Formula

(38)

where

(39)

Todd (1949) gives a table of decompositions of

for

. Conway and Guy (1996) give a similar table interms of Stérmer Numbers.

Arndt and Gosper give the remarkable inverse tangent identity

(40)

See also Inverse Cotangent, Tangent

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Inverse Circular Functions.'' §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79-83, 1972.

Acton, F. S. ``The Arctangent.'' In Numerical Methods that Work, upd. and rev. Washington, DC: Math. Assoc. Amer., pp. 6-10, 1990.

Arndt, J. ``Completely Useless Formulas.'' http://www.jjj.de/hfloat/hfloatpage.html#formulas.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Item 137, Feb. 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142-143, 1987.

Castellanos, D. ``Rapidly Converging Expansions with Fibonacci Coefficients.'' Fib. Quart. 24, 70-82, 1986.

Castellanos, D. ``The Ubiquitous Pi. Part I.'' Math. Mag. 61, 67-98, 1988.

Conway, J. H. and Guy, R. K. ``Stérmer's Numbers.'' The Book of Numbers. New York: Springer-Verlag, pp. 245-248, 1996.

Todd, J. ``A Problem on Arc Tangent Relations.'' Amer. Math. Monthly 56, 517-528, 1949.

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。