ArcCos. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation

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1 ArcCos Notations Traditional name Inverse cosine Traditional notation cos Mathematica StandardForm notation ArcCos Primary definition log The function cos can also be defined by the formula cos sin. The function cos can be defined also as the inverse function for cosw: w cos if and only if cosw. Specific values Values at fixed points cos

2 cos cos cos cos cos cos

3 cos cos cos cos cos cos cos cos log cos log Values at infinities cos cos cos cos

4 cos General characteristics Domain and analyticity cos is an analytical function of, which is defined over the whole complex -plane cos Symmetries and periodicities Mirror symmetry cos cos ;,, Periodicity No periodicity Poles and essential singularities The function cos does not have poles and essential singularities ing cos Branch points The function cos has three branch points: ±, cos,, cos, cos, cos, log Branch cuts The function cos is a single-valued function on the -plane cut along the intervals, and,. The function cos is continuous from above on the interval, and from below on the interval, cos,,,,,

5 lim Ε0 cos x Ε cos x ; x lim Ε0 cos x Ε cos x ; x lim Ε0 cos x Ε cos x ; x lim Ε0 cos x Ε cos x ; x Analytic continuations The analytic continuation of cos has infinitely many sheets; the values of cos cos cos k ; k. are Series representations Generalied power series Expansions at 0 cos ; k k ; k k k F, ; ; ;, cos O ; 0 Expansions at cos 60 ; cos k 0 k ; k k k k cos F, ; ;

6 cos O ; Expansions at cos 60 ; cos k 0 k ; k k k k cos F, ; ; cos O ; Expansions at log 6 ; log k ; k k k k log F,, ;, ; ;, cos log O ; Residue representations cos j s s res s s j ; s s res s s s j 0 s j ;

7 7 Integral representations On the real axis Of the direct function cos t t Contour integral representations s s s s s ; Arg s s s s s ; Arg Γ s s s s s ; 0 Γ Γ Γ s Γ s s Arg s s ; 0 Γ Arg Continued fraction representations ;,, k k k ;,,, k

8 8 Differential equations Ordinary linear differential equations and wronskians For the direct function itself w w 0 ; w cos w0 w w w 0 ; w c c cos W, cos w ; w cos w0 Transformations Transformations and argument simplifications Argument involving basic arithmetic operations cos cos cos a b c m b c m b m m c cos a b m m c ; m cos cos cos cos cos cos cos cos cos cos

9 cos cos cos ;,, Products, sums, and powers of the direct function Sums of the direct function cos x cos y sgnx y cos x y x y sgnx y ; x y x y cos x cos y sgnx y sgnx y cos x y x y ; x y x y cos x cos y sgnx y cos x y x y ; x y x y cos x cos y sgnx y cos x y x y ; x y x y Related transformations Sums involving the direct function cos x sin y sgnx y cos x y x y ; x y x y cos x sin y sgnx y cos x y x y ; x y x y cos x sin y sgnx y cos x y x y sgnx y ; x y x y cos x sin y sgnx y sgnx y cos x y x y ; x y x y Identities Functional identities cos w w cosw w ; w cos

10 0 Complex characteristics Real part Recos x y cos x y Recos x y tan x y x y cos sin tan x y, x y x y x y Imaginary part Imcos x y sgny logx X ; X x y x y ; x y, x y, tan x y, x y y, x x y x y, x y, Imcos x y log y x y x y cos tan x y, x y sin tan x y, x y x y x y Absolute value cos x y tan x y x y cos tan x y, x y y, sin tan x y, x y x y x y x ^ x log y x y x y cos tan x y, x y sin tan x y, x y x y x y x

11 Argument Argcos x y tan tan x y x y cos tan x y, x y y, sin tan x y, x y x y x y x, log y x y x y cos tan x y, x y Conjugate value sin tan x y, x y x y x y cos x y tan x y x y cos tan x y, x y y, sin tan x y, x y x y x y x log y x y x y cos tan x y, x y sin tan x y, x y x y x y x ^ x Differentiation Low-order differentiation cos cos Symbolic differentiation n cos n F n n, n, ; Fractional integro-differentiation, n ; ; n Α cos Α Α Α Α Α F, Α, ;, Α ;

12 Integration Indefinite integration For the direct function itself cos cos cos log cos Li cos cos cos E sin Fsin Α cos cos Α Α Α Α Α F Α, ; Α ; cos b a cos b a b sin b a b a a a cos b a a cos b a b sin b a b a b a a cos a b cos b a sin b tan b b tan cos b a cos b a sin b log cos ba b b cos b a sin b Li b b cos ba Li b b cos ba log b b cos ba cos Sicos cos n cos n cos n n, cos cos n cos n n, cos cos n n cos n cos n n, cos n cos n cos n n, cos

13 Definite integration For the direct function itself t cos tt cos t t 0 t a t a cos tt ; Rea 0 a a Involving the direct function logt cos tt log 0 Representations through more general functions Through hypergeometric functions Involving F F, ; ; cos F, ; ; cos F, ; ; Involving p F q log F,, ;, ; ;, Through Meijer G Classical cases for the direct function itself cos G,,, 0,

14 cos G,,,, cos G,,,, cos G,, cos G,,,, 0 ; Im 0,, 0 ; Im cos G,,, 0, Classical cases involving algebraic functions in the arguments cos G,, cos G,, cos G,, cos G,,,,,,, 0, 0,, 0 ;, 0 Classical cases involving unit step Θ,,,,, 0 ;, 0, 0,, 0 ;, Θ cos,0 G,, 0, ;, Θ cos 0, G,, 0, 0

15 5 Generalied cases for the direct function itself cos G,,, cos G,,,,,,, 0 Generalied cases involving algebraic functions in the arguments cos G,,,,,, cos G,,,, 0, cos G,,, , 0 ;, 0,, 0,,, cos G,,, 0, Generalied cases involving unit step Θ,, 0 ;, 0,, Θ cos,0 G,,, 0, Θ cos 0, G,, Through other functions Involving inverse Jacobi functions cos cd cos cn cos dc ds 0, 0, 0

16 cos nc ns sd sn 0 Involving some hypergeometric-type functions , , Representations through equivalent functions With inverse function cos cos ; 0 Re Re 0 Im 0 Re Im cos cos ; Re 0 Re 0 Im 0 Re Im cos cos k k k ; k Re k Re k Im 0 Re k Im 0 k cos cos Re Re Re Re ΘIm Re coscos With related functions Involving log log cos log ; Re Im 0 Re 0

17 cos log cos log log cos log Involving sin sin cos sin cos sin cos sin cos sin cos sin cos sin Involving tan cos tan ;, cos tan ; cos tan ;,, cos tan

18 tan cos tan ;,, cos r c r c r r c c r r c c tan r c r c r c a a a a a a tan a cos tan tan cot Involving cot cos cot ; cos cot a c a c cot a c a c a a c a c c a c Involving csc cos csc csc

19 cos csc csc cos csc ; Re cos csc cos r c c r r c c c r r c r csc c r a a a a a csc a cos csc Involving sec cos sec cos sec sec ;, 0 Involving sinh cos sinh cos sinh cos sinh

20 cos sinh cos a c a c sinh a c a c Involving cosh cos cosh cos cosh cos cosh ; Im cos cosh ; Im 0 Involving tanh cos tanh ;,, tanh cos tanh ;,, cos tanh cos r c r c r r c c r r c c r c tanh r c r c a a a a a a a tanh a a cos tanh

21 Involving coth cos coth cos coth coth Involving csch cos csch cos csch ; Re cos csch ; Re Im cos csch cos r c c r r c r c c r c r r c a a a a a csch a csch csch csch c r csch c r

22 Involving sech sech Inequalities cos x 0 ; x x cos x ; x x Zeros cos 0 ; History J. Herschel (8) introduced the notation cos The function cos is often encountered in mathematics and the natural sciences.

23 Copyright This document was downloaded from functions.wolfram.com, a comprehensive online compendium of formulas involving the special functions of mathematics. For a key to the notations used here, see Please cite this document by referring to the functions.wolfram.com page from which it was downloaded, for example: To refer to a particular formula, cite functions.wolfram.com followed by the citation number. e.g.: This document is currently in a preliminary form. If you have comments or suggestions, please comments@functions.wolfram.com. 00, Wolfram Research, Inc.

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