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ArcCos

Notationsÿ

Traditional nameÿ

Inverse cosine

Traditional notationÿ

cos-1 HzLMathematica StandardForm notationÿ

ArcCos@zDPrimary definitionÿ

01.13.02.0001.01

cos-1 HzL �Π��2

+ ä logIä z +�!!!!!!!!!!!!!

1 - z2 MThe function cos-1 HzL can also be defined by the formula cos-1 HzL � Π��2 - sin-1 HzL .

The function cos-1 HzL can be defined also as the inverse function for cosHwL : w = cos-1 HzL if and only ifcosHwL = z .

Specific valuesÿ

Values at fixed pointsÿ

01.13.03.0001.01

cos-1 H0L �Π��2

01.13.03.0002.01

cos-1ikjjjjj �!!!!3 - 1

��2 �!!!!2

y{zzzzz �5 Π��12

01.13.03.0003.01

cos-1ikjjjjj-

�!!!!3 - 1��

2 �!!!!2


01.13.03.0004.01

cos-1ikjjjjj �!!!!5 - 1

��4


01.13.03.0005.01

cos-1ikjjjjj-

�!!!!5 - 1��

4


01.13.03.0006.01

cos-1ikjjjjjjjj "#################

2 -�!!!!2

��2

y{zzzzzzzz �3 Π��8

01.13.03.0007.01

cos-1ikjjjjjjjj-

"#################2 -

�!!!!2��

2


01.13.03.0008.01

cos-1 ikjj 1��2

y{zz �Π��3

01.13.03.0009.01

cos-1 ikjj-1��2

y{zz �2 Π��3

01.13.03.0010.01

cos-1ikjjjjjjjj 1

��2

$%%%%%%%%%%%%%%%%%%%%5 -�!!!!5

��2


01.13.03.0011.01

cos-1ikjjjjjjjj-

1��2

$%%%%%%%%%%%%%%%%%%%%5 -�!!!!5

��2


01.13.03.0012.01

cos-1ikjjjjj �!!!!2

��2

y{zzzzz �Π��4

01.13.03.0013.01

cos-1ikjjjjj-

�!!!!2��

2


01.13.03.0014.01

cos-1ikjjjjj �!!!!5 + 1

��4


01.13.03.0015.01

cos-1ikjjjjj-

�!!!!5 + 1��

4


01.13.03.0016.01

cos-1ikjjjjj �!!!!3

��2


http://functions.wolfram.com 2

01.13.03.0017.01

cos-1ikjjjjj-

�!!!!3��

2


01.13.03.0018.01

cos-1ikjjjjjjjj "#################

2 +�!!!!2

��2

y{zzzzzzzz �Π��8

01.13.03.0019.01

cos-1ikjjjjjjjj-

"#################2 +

�!!!!2��

2


01.13.03.0020.01

cos-1ikjjjjjjjj 1

��2

$%%%%%%%%%%%%%%%%%%%%5 +�!!!!5

��2

y{zzzzzzzz �Π

��10

01.13.03.0021.01

cos-1ikjjjjjjjj-

1��2

$%%%%%%%%%%%%%%%%%%%%5 +�!!!!5

��2


01.13.03.0022.01

cos-1ikjjjjj 1 +

�!!!!3��

2 �!!!!2

y{zzzzz �Π

��12

01.13.03.0023.01

cos-1ikjjjjj-

1 +�!!!!3

��2 �!!!!2


01.13.03.0024.01

cos-1 H1L � 0

01.13.03.0025.01

cos-1 H-1L � Π

01.13.03.0026.01

cos-1 HäL �Π��2

+ ä logI�!!!!2 - 1M01.13.03.0027.01

cos-1 H-äL �Π��2

+ ä logI1 +�!!!!2 M

Values at infinitiesÿ

01.13.03.0028.01

cos-1 H¥L � ä ¥

01.13.03.0029.01

cos-1 H-¥L � -ä ¥

01.13.03.0030.01

cos-1 Hä ¥L � -ä ¥

01.13.03.0031.01

cos-1 H-ä ¥L � ä ¥


01.13.03.0032.01

cos-1 H¥� L � ¥�

General characteristicsÿ

Domain and analyticityÿ

cos-1 HzL is an analytical function of z , which is defined over the whole complex z-plane.

01.13.04.0001.01

z�cos-1 HzL � Â�Â

Symmetries and periodicitiesÿ

Mirror symmetryÿ

01.13.04.0002.01

cos-1 Hz�L � cos-1 HzL�� ; z Ï H-¥, -1L ß z Ï H1, ¥LPeriodicityÿ

No periodicity

Poles and essential singularitiesÿ

The function cos-1 HzL does not have poles and essential singularities.

01.13.04.0003.01

Singz Hcos-1 HzLL � 8<Branch pointsÿ

The function cos-1 HzL has three branch points: z = ± 1, z = ¥� .

01.13.04.0004.01

BPz Hcos-1 HzLL � 81, -1, ¥� <01.13.04.0005.01

Rz Hcos-1 HzL, 1L � 2

01.13.04.0006.01

Rz Hcos-1 HzL, -1L � 2

01.13.04.0007.01

Rz Hcos-1 HzL, ¥� L � log

Branch cutsÿ

The function cos-1 HzL is a single-valued function on the z-plane cut along the intervals H-¥, -1D and @1, ¥L . The function cos-1 HzL is continuous from above on the interval H-¥, -1D and from below on the interval @1, ¥L .

01.13.04.0008.01

BCz Hcos-1 HzLL � 88H-¥, -1D, -ä<, 8@1, ¥L, ä<<


01.13.04.0009.01

limΕ®+0

cos-1 Hx + ä ΕL � cos-1 HxL �; x < -1

01.13.04.0010.01

limΕ®+0

cos-1 Hx - ä ΕL � cos-1 H-xL + Π �; x < -1

01.13.04.0011.01

limΕ®+0

cos-1 Hx - ä ΕL � cos-1 HxL �; x > 1

01.13.04.0012.01

limΕ®+0

cos-1 Hx + ä ΕL � -cos-1 HxL �; x > 1

Analytic continuationsÿ

The analytic continuation of cos-1 has infinitely many sheets; the values of cos� -1 arecos� -1 HzL = cos-1 HzL + 2 k Π �; k Î Ù .

Series representationsÿ

Generalized power seriesÿ

Expansions at z � 0ÿ

01.13.06.0001.01


- z -z3

��6

-3 z5

��40

- ¼ �; z¤ < 1

01.13.06.0002.01


- ãk=0

¥ H 1��2 Lk

z2 k+1

��H2 k + 1L k !�; z¤ < 1

01.13.06.0003.01


- z 2 F1ikjj 1

��2

,1��2

;3��2

; z2 y{zz �; z Ï H-1, 1L01.13.06.0004.01

cos-1 HzL µΠ��2

- z + OHz3 L �; Hz ® 0L Expansions at z � 1ÿ

01.13.06.0005.01

cos-1 HzL ��!!!!2 �!!!!!!!!!!!1 - z ikjj1 +

1 - z��

12+

3��160

H1 - zL2 + ¼y{zz �; z - 1¤ < 1

01.13.06.0006.01

cos-1 HzL ��!!!!2 �!!!!!!!!!!!1 - z ã

k=0

¥ H 1��2 Lk

H1 - zLk

��2k H2 k + 1L k !

�; z - 1¤ < 1

01.13.06.0007.01

cos-1 HzL ��!!!!2 �!!!!!!!!!!!1 - z 2 F1

ikjj 1��2

,1��2

;3��2

;1 - z��

2y{zz


01.13.06.0008.01

cos-1 HzL µ�!!!!2 �!!!!!!!!!!!1 - z H1 + OHz - 1LL �; Hz ® 1L

Expansions at z � -1ÿ

01.13.06.0009.01

cos-1 HzL � Π -�!!!!2 �!!!!!!!!!!!1 + z ikjj1 +

1 + z��

12+

3��160

H1 + zL2 + ¼y{zz �; z + 1¤ < 1

01.13.06.0010.01

cos-1 HzL � Π -�!!!!2 �!!!!!!!!!!!z + 1 ã

k=0

¥ H 1��2 Lk

Hz + 1Lk

��2k H2 k + 1L k !

�; z + 1¤ < 1

01.13.06.0011.01

cos-1 HzL � Π -�!!!!2 �!!!!!!!!!!!z + 1 2 F1

ikjj 1��2

,1��2

;3��2

;z + 1��

2y{zz

01.13.06.0012.01

cos-1 HzL µ Π -�!!!!2 �!!!!!!!!!!!z + 1 H1 + OHz + 1LL �; Hz ® -1L

Expansions at z � ¥ÿ

01.13.06.0013.01


-z

��2

�!!!!!!!!!-z2

ikjjlogH-4 z2 L -1

��2 z2

-3

��16 z4

- ¼y{zz �; z¤ > 1

01.13.06.0014.01


-z

��2

�!!!!!!!!!-z2

ikjjjjjjjjlogH-4 z2 L - ã

k=1

¥ H 1��2 Lk

z-2 k

��k k !

y{zzzzzzzz �; z¤ > 1

01.13.06.0015.01


-z

��2

�!!!!!!!!!-z2

ikjjlogH-4 z2 L -1

��2 z2

3 F2ikjj 3

��2

, 1, 1; 2, 2;1

��z2

y{zzy{zz �; z Ï H-1, 1L01.13.06.0016.01

cos-1 HzL µΠ��2

-z logH-4 z2 L��

2�!!!!!!!!!

-z2+

1��4 z

�!!!!!!!!!-z2

ikjj1 + Oikjj 1��z2

y{zzy{zz �; H z¤ ® ¥LResidue representationsÿ

01.13.06.0017.01

cos-1 HzL �1

��2 z �!!!!

Πã

j=1

¥

ress

ikjjjjjjj GHs + 1L GH-s - 1��2 L2 H-z2 L-s

��GH 1��2 - sL y{zzzzzzz H- jL +

Π��2

�; z¤ < 1

01.13.06.0018.01

cos-1 HzL �z

��2 �!!!!

Π ã

j=0

¥

ress

ikjjjjjjj GHsL GH 1��2 - sL2 H-z2 L-s

��GH 3��2 - sL y{zzzzzzz ikjj 1

��2

+ jy{zz +Π��2

�; z¤ > 1


Integral representationsÿ

On the real axisÿ

Of the direct functionÿ

01.13.07.0001.01

cos-1 HzL � àz

1 1��!!!!!!!!!!!!

1 - t2 â t

Contour integral representationsÿ

01.13.07.0002.01


-z

��I2 �!!!!Π M 2 Π ä

áL

GHsL GH 1��2 - sL2

��GH 3��2 - sL H-z2 L-s

â s �; ArgH-z2 L¤ < Π

01.13.07.0003.01


-z

��I2 �!!!!Π M 2 Π ä

àL

GHsL Gikjjs +1��2

y{zz Gikjj 1��2

- sy{zz2 H1 - z2 L-s â s �; ArgH1 - z2 L¤ < Π

01.13.07.0004.01


-z

��I2 �!!!!Π M 2 Π ä

áΓ-ä ¥

Γ+ä ¥GHsL GH 1��2 - sL2

��GH 3��2 - sL H-z2 L-s

â s �; 0 < Γ <1��2

í ArgH-z2 L¤ < Π

01.13.07.0005.01


-z

��I2 �!!!!Π M 2 Π ä

àΓ-ä ¥

Γ+ä ¥

GHsL Gikjjs +1��2

y{zz Gikjj 1��2

- sy{zz2 H1 - z2 L-s â s �; 0 < Γ <

1��2

í ArgH1 - z2 L¤ < Π

Continued fraction representationsÿ

01.13.10.0001.01


-z

�!!!!!!!!!!!!!1 - z2

��

1 -1 ´ 2 z2

��

3 -1 ´ 2 z2

��

5 -3 ´ 4 z2

��

7 -3 ´ 4 z2

��

9 -5 ´ 6 z2

��11 - ¼

�; z Ï H-¥, -1L ß z Ï H1, ¥L

01.13.10.0002.01


-z

�!!!!!!!!!!!!!1 - z2

��1 + Kk H-2 H2 d k+1��2 t - 1L d k+1��2 t z2 , 2 k + 1L

1

¥ �; z Ï H-¥, -1L ß z Ï H1, ¥L


Differential equationsÿ

Ordinary linear differential equations and wronskiansÿ

For the direct function itselfÿ

01.13.13.0001.01H1 - z2 L w¢¢ HzL - z w¢ HzL � 0 �; wHzL � cos-1 HzL í wH0L �Π��2

í w¢ H0L � -1

01.13.13.0002.01H1 - z2 L w¢¢ HzL - z w¢ HzL � 0 �; wHzL � c1 + c2 cos-1 HzL01.13.13.0003.01

Wz H1, cos-1 HzLL � -1

��!!!!!!!!!!!!!1 - z2

01.13.13.0004.01�!!!!!!!!!!!!!1 - z2 w¢ HzL � -1 �; wHzL � cos-1 HzL í wH0L �

Π��2

Transformationsÿ

Transformations and argument simplificationsÿ

Argument involving basic arithmetic operationsÿ

01.13.16.0001.01

cos-1 H-zL � Π - cos-1 HzL01.13.16.0002.01

cos-1 Ha Hb zc Lm L �Π��2

-Hb zc Lm

��bm zm c

J Π��2

- cos-1 Ha bm zm c LN �; 2 m Î Ù

01.13.16.0003.01

cos-1 H1 - 2 z2 L �2

�!!!!!z2

��z

J Π��2

- cos-1 HzLN01.13.16.0004.01

cos-1 H2 z2 - 1L �2

�!!!!!z2

��z

cos-1 HzL + Πikjjjjjjj-$%%%%%%%%1

��z2

z - $%%%%%%ä��z

�!!!!!!!!!-ä z + $%%%%%%%%%-

ä��z

�!!!!!!ä z + 1

y{zzzzzzz01.13.16.0005.01

cos-1ikjjjjjjj$%%%%%%%%%%%%%z + 1

��2

y{zzzzzzz �1��2

cos-1 HzL01.13.16.0006.01

cos-1 I�!!!!!z2 M �

Π��2

-�!!!!!

z2��

z J Π

��2

- cos-1 HzLN01.13.16.0007.01

cos-1 I�!!!!!!!!!!!!!1 - z2 M �

�!!!!!z2

��z

J Π��2

- cos-1 HzLN


01.13.16.0008.01

cos-1ikjjjjjjj$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%1

��2

I1 -�!!!!!!!!!!!!!

1 - z2 M y{zzzzzzz �Π��2

-�!!!!!

z2��

2 z J Π

��2

- cos-1 HzLN01.13.16.0009.01

cos-1 I2 z�!!!!!!!!!!!!!

1 - z2 M � 2 cos-1 HzL -Π��2

�; z Ïikjjjj-¥, -

1��!!!!2

y{zzzz í z Ïikjjjj 1

��!!!!2, ¥

y{zzzzProducts, sums, and powers of the direct functionÿ

Sums of the direct functionÿ

01.13.16.0010.01

cos-1 HxL + cos-1 HyL � Π H1 - sgnHx + yLL + cos-1 Ix y -�!!!!!!!!!!!!!

1 - x2 �!!!!!!!!!!!!!1 - y2 M sgnHx + yL �; -1 < x < 1 Þ -1 < y < 1 Þ x y > 0

01.13.16.0011.01

cos-1 HxL + cos-1 HyL � Π HsgnHx + yL + 1L - sgnHx + yL cos-1 Ix y -�!!!!!!!!!!!!!

1 - x2 �!!!!!!!!!!!!!1 - y2 M �; x < -1 ß y > 1 Þ x > 1 ß y < -1

01.13.16.0012.01

cos-1 HxL - cos-1 HyL � -sgnHx - yL cos-1 Ix y +�!!!!!!!!!!!!!

1 - x2 �!!!!!!!!!!!!!1 - y2 M �; -1 < x < 1 Þ -1 < y < 1 Þ x y < 0

01.13.16.0013.01

cos-1 HxL - cos-1 HyL � sgnHx - yL cos-1 Ix y +�!!!!!!!!!!!!!

1 - x2 �!!!!!!!!!!!!!1 - y2 M �; x > 1 ß y > 1 Þ x < -1 ß y < -1

Related transformationsÿ

Sums involving the direct functionÿ

01.13.16.0014.01

cos-1 HxL + sin-1 HyL �Π��2

- sgnHx - yL cos-1 Ix y +�!!!!!!!!!!!!!

1 - x2 �!!!!!!!!!!!!!1 - y2 M �; -1 < x < 1 Þ -1 < y < 1 Þ x y < 0

01.13.16.0015.01

cos-1 HxL + sin-1 HyL �Π��2

- sgnHx - yL cos-1 Ix y +�!!!!!!!!!!!!!

1 - x2 �!!!!!!!!!!!!!1 - y2 M �; -1 < x < 1 Þ -1 < y < 1 Þ x y < 0

01.13.16.0016.01

cos-1 HxL - sin-1 HyL � Π ikjj 1��2

- sgnHx + yLy{zz + cos-1 Ix y -�!!!!!!!!!!!!!

1 - x2 �!!!!!!!!!!!!!1 - y2 M sgnHx + yL �; -1 < x < 1 Þ -1 < y < 1 Þ x y > 0

01.13.16.0017.01

cos-1 HxL - sin-1 HyL � Π ikjjsgnHx + yL +1��2

y{zz - sgnHx + yL cos-1 Ix y -�!!!!!!!!!!!!!

1 - x2 �!!!!!!!!!!!!!1 - y2 M �; x < -1 ß y > 1 Þ x > 1 ß y < -1

Identitiesÿ

Functional identitiesÿ

01.13.17.0001.01

cos2 HwHz1 L + wHz2 LL - 2 z1 z2 cosHwHz1 L + wHz2 LL + z12 + z2

2 � 1 �; wHzL � cos-1 HzL


Complex characteristicsÿ

Real partÿ

01.13.19.0001.01

ReHcos-1 Hx + ä yLL � cos-1 ikjj 1��2

"##########################Hx + 1L2 + y2 -1��2

"##########################Hx - 1L2 + y2 y{zz �; x + ä y Ï H-¥, -1L ß x + ä y Ï H1, ¥L01.13.19.0002.01

ReHcos-1 Hx + ä yLL �1��2

ikjjΠ - 2 tan-1 ikjj"###################################################4 x2 y2 + H-x2 + y2 + 1L24cosikjj 1

��2

tan-1 H-x2 + y2 + 1, -2 x yLy{zz - y,

sinikjj 1��2

tan-1 H-x2 + y2 + 1, -2 x yLy{zz "###################################################4 x2 y2 + H-x2 + y2 + 1L24+ xy{zzy{zz

Imaginary partÿ

01.13.19.0003.01

ImHcos-1 Hx + ä yLL � -sgnHyL logIX +�!!!!!!!!!!!!!!

X2 - 1 M �;X �

1��2

"##########################Hx - 1L2 + y2 +1��2

"##########################Hx + 1L2 + y2 í x + ä y Ï H-¥, -1L í x + ä y Ï H1, ¥L01.13.19.0004.01

ImHcos-1 Hx + ä yLL � logikjjjj-ikjjjjikjjy -

"##########################################################x4 + 2 Hy2 - 1L x2 + Hy2 + 1L24cosikjj 1

��2

tan-1 H-x2 + y2 + 1, -2 x yLy{zzy{zz2

+ikjjsinikjj 1��2

tan-1 H-x2 + y2 + 1, -2 x yLy{zz "##########################################################x4 + 2 Hy2 - 1L x2 + Hy2 + 1L24+ xy{zz2 y{zzzzy{zzzz

Absolute valueÿ

01.13.19.0005.01 cos-1 Hx + ä yL¤ �-ikjjjj 1��4

ikjjΠ - 2 tan-1 ikjj"###################################################4 x2 y2 + H-x2 + y2 + 1L24cosikjj 1

��2

tan-1 H-x2 + y2 + 1, -2 x yLy{zz - y, sinikjj 1��2

tan-1 H-x2 + y2 + 1, -2 x yLy{zz"###################################################4 x2 y2 + H-x2 + y2 + 1L24+ xy{zzy{zz^2 +

log2 ikjjjj-ikjjjjikjjy -"##########################################################

x4 + 2 Hy2 - 1L x2 + Hy2 + 1L24cosikjj 1

��2



tan-1 H-x2 + y2 + 1, -2 x yLy{zz "##########################################################x4 + 2 Hy2 - 1L x2 + Hy2 + 1L24+ xy{zz2 y{zzzzy{zzzzy{zzzz


Argumentÿ

01.13.19.0006.01

ArgHcos-1 Hx + ä yLL � tan-1 ikjjjjΠ - 2 tan-1 ikjj"###################################################4 x2 y2 + H-x2 + y2 + 1L24cosikjj 1

��2

tan-1 H-x2 + y2 + 1, -2 x yLy{zz - y,

sinikjj 1��2

tan-1 H-x2 + y2 + 1, -2 x yLy{zz "###################################################4 x2 y2 + H-x2 + y2 + 1L24+ xy{zz,

2 logikjjjj-ikjjjjikjjy -


��2



tan-1 H-x2 + y2 + 1, -2 x yLy{zz "##########################################################x4 + 2 Hy2 - 1L x2 + Hy2 + 1L24+ xy{zz2 y{zzzzy{zzzzy{zzzz

Conjugate valueÿ

01.13.19.0007.01

cos-1 Hx + ä yL��

1��2

ikjjjj-2 tan-1 ikjj"##########################################################x4 + 2 Hy2 - 1L x2 + Hy2 + 1L24cosikjj 1

��2

tan-1 H-x2 + y2 + 1, -2 x yLy{zz - y,

sinikjj 1��2

tan-1 H-x2 + y2 + 1, -2 x yLy{zz "##########################################################x4 + 2 Hy2 - 1L x2 + Hy2 + 1L24+ xy{zz -

2 ä logikjjjj-ikjjjjikjjy -


��2



tan-1 H-x2 + y2 + 1, -2 x yLy{zz "##########################################################x4 + 2 Hy2 - 1L x2 + Hy2 + 1L24+ xy{zz^2

y{zzzzy{zzzz + Πy{zzzz

Differentiationÿ

Low-order differentiationÿ

01.13.20.0001.01

¶cos-1 HzL��

¶z� -

1��!!!!!!!!!!!!!

1 - z2

01.13.20.0002.01

¶2 cos-1 HzL��

¶z2� -

z��H1 - z2 L3�2

Symbolic differentiation ÿ

01.13.20.0003.01

¶n cos-1 HzL��

¶zn� -

�!!!!Π z1-n

��21-n 3 F

�2

ikjj 1��2

,1��2

, 1; 1 -n��2

,3 - n��

2; z2 y{zz �; n Î Í+

Fractional integro-differentiationÿ

01.13.20.0004.01

¶Α cos-1 HzL��

¶zΑ�

Π z-Α

��2 GH1 - ΑL - 2Α-1 �!!!!

Π z1-Α3 F

�2

ikjj 1��2

,1��2

, 1; 1 -Α��2

,3��2

-Α��2

; z2 y{zz


Integrationÿ

Indefinite integrationÿ


01.13.21.0001.01à cos-1 HzL â z � z cos-1 HzL -�!!!!!!!!!!!!!

1 - z2

01.13.21.0002.01à cos-1 HzL��

z â z � -

ä��2

cos-1 HzL2+ logI1 + ã2 ä cos-1 HzL M cos-1 HzL -

ä��2

Li2 I-ã2 ä cos-1 HzL M01.13.21.0003.01à cos-1 HzL

��!!!z â z � 2 �!!!z ikjjjjjjjjcos-1 HzL +�!!!!!!!!!!!z + 1

��"###########z��2 z-2�!!!!!!!!!!!!!

1 - z2

ikjj2 Eikjjsin-1 I�!!!!!!!!!!!z + 1 M ÄÄÄÄÄÄÄÄÄ 1��2

y{zz - Fikjjsin-1 I�!!!!!!!!!!!z + 1 M ÄÄÄÄÄÄÄÄÄ 1��2

y{zzy{zzy{zzzzzzzz01.13.21.0004.01à zΑ-1 cos-1 HzL â z �

cos-1 HzL zΑ

��Α

+zΑ+1

��Α HΑ + 1L 2 F1

ikjj Α + 1��

2,

1��2

;Α + 3��

2; z2 y{zz

01.13.21.0005.01à cos-1 Hb + a zL â z � z cos-1 Hb + a zL -b sin-1 Hb + a zL��

a-

"###########################1 - Hb + a zL2

��a

01.13.21.0006.01à z cos-1 Hb + a zL â z �2 a2 cos-1 Hb + a zL z2 + H2 b2 + 1L sin-1 Hb + a zL + H3 b - a zL "###########################1 - Hb + a zL2

��4 a2

01.13.21.0007.01à cos-1 Ha z + bL��

z â z � -

ä��2

cos-1 Hb + a zL2- 4 ä sin-1 ikjjjjj �!!!!!!!!!!!1 - b

��!!!!2

y{zzzzz tan-1 ikjjjj b + 1��!!!!!!!!!!!!!

b2 - 1tanikjj 1

��2

cos-1 Hb + a zLy{zzy{zzzz +ikjjjjjcos-1 Hb + a zL - 2 sin-1 ikjjjjj �!!!!!!!!!!!1 - b��!!!!2

y{zzzzzy{zzzzz logIãä cos-1 Hb+a zL I�!!!!!!!!!!!!!b2 - 1 - bM + 1M +ikjjjjjcos-1 Hb + a zL + 2 sin-1 ikjjjjj �!!!!!!!!!!!1 - b

��!!!!2

y{zzzzzy{zzzzz logI1 - Ib +�!!!!!!!!!!!!!

b2 - 1 M ãä cos-1 Hb+a zL M -

ä ILi2 IIb +�!!!!!!!!!!!!!

b2 - 1 M ãä cos-1 Hb+a zL M + Li2 I-I�!!!!!!!!!!!!!b2 - 1 - bM ãä cos-1 Hb+a zL MM

01.13.21.0008.01à 1��cos-1 HzL â z � -SiHcos-1 HzLL

01.13.21.0009.01à cos-1 HzLn â z �

1��2

ä IHä cos-1 HzLL-n-1cos-1 HzLn+1

GHn + 1, ä cos-1 HzLL - H-ä cos-1 HzLL-n-1cos-1 HzLn+1

GHn + 1, -ä cos-1 HzLLM01.13.21.0010.01à z cos-1 HzLn

â z �

1��4

ä I2-n-1 Hä cos-1 HzLL-n-1cos-1 HzLn+1

GHn + 1, 2 ä cos-1 HzLL - 2-n-1 H-ä cos-1 HzLL-n-1cos-1 HzLn+1

GHn + 1, -2 ä cos-1 HzLLM


Definite integrationÿ


01.13.21.0011.01à0

1

t cos-1 HtL â t �Π��8

01.13.21.0012.01à0

1 cos-1 HtL��!!!t â t �

�!!!!Π GH 3��4 L

��GH 5��4 L

01.13.21.0013.01à0

1

ta cos-1 HtL â t �

�!!!!Π GH a+2��2 L

��Ha + 1L2 GH a+1��2 L �; ReHaL > -1

Involving the direct functionÿ

01.13.21.0014.01à0

1

logHtL cos-1 HtL â t � logH2L - 2

Representations through more general functionsÿ

Through hypergeometric functionsÿ

Involving 2 F1 ÿ

01.13.26.0001.01


- z 2 F1ikjj 1

��2

,1��2

;3��2

; z2 y{zz01.13.26.0002.01

cos-1 HzL ��!!!!2 �!!!!!!!!!!!1 - z 2 F1

ikjj 1��2

,1��2

;3��2

;1 - z��

2y{zz

01.13.26.0003.01

cos-1 HzL � Π -�!!!!2 �!!!!!!!!!!!z + 1 2 F1

ikjj 1��2

,1��2

;3��2

;z + 1��

2y{zz

Involving p Fq ÿ

01.13.26.0004.01


-z

��2

�!!!!!!!!!-z2

ikjjlogH-4 z2 L -1

��2 z2

3 F2ikjj 3

��2

, 1, 1; 2, 2;1

��z2

y{zzy{zz �; z Ï H-1, 1LThrough Meijer Gÿ

Classical cases for the direct function itselfÿ

01.13.26.0005.01


-z

��2 �!!!!

Π G2,2

1,2ikjjjjjj-z2

ÄÄÄÄÄÄÄÄÄÄÄÄÄ 1��2 , 1��2

0, - 1��2

y{zzzzzz


01.13.26.0006.01


+1

��2 �!!!!

Π zG2,2

1,2ikjjjjjj-z2


1, 1��2

y{zzzzzz01.13.26.0007.01


+�!!!!!!!!!

-z2��2 z �!!!!

Π G2,2

1,2 ikjjjjj-z2ÄÄÄÄÄÄÄÄÄÄÄ 1, 1

1��2 , 0y{zzzzz

01.13.26.0008.01


-ä

��2 �!!!!

Π G2,2


1��2 , 0y{zzzzz �; Im HzL > 0

01.13.26.0009.01


-ä

��2 �!!!!

Π G2,2


1��2 , 0y{zzzzz �; Im HzL > 0

01.13.26.0010.01

cos-1 I�!!!z M �Π��2

-�!!!z

��2 �!!!!

Π G2,2

1,2ikjjjjjj-z


0, - 1��2

y{zzzzzzClassical cases involving algebraic functions in the argumentsÿ

01.13.26.0011.01

cos-1 I�!!!!!!!!!!!z + 1 -�!!!z M �

1��2 �!!!!!!!2 Π

G3,32,2

ikjjjjjjz

ÄÄÄÄÄÄÄÄÄÄÄÄÄ 1��2 , 1, 11��4 , 3��4 , 0

y{zzzzzz01.13.26.0012.01

cos-1 ikjjjjj �!!!!!!!!!!!z + 1 - 1

��!!!z y{zzzzz �1

��2 �!!!!!!!2 Π

G3,32,2

ikjjjjjjz

ÄÄÄÄÄÄÄÄÄÄÄÄÄ 1��4 , 3��4 , 1

0, 1��2 , 0

y{zzzzzz �; z Ï H-¥, 0L01.13.26.0013.01

cos-1 ikjjjj 1

��!!!!!!!!!!!z + 1 +�!!!z y{zzzz �

1��2 �!!!!!!!2 Π

G3,32,2

ikjjjjjjz

ÄÄÄÄÄÄÄÄÄÄÄÄÄ 1��2 , 1, 11��4 , 3��4 , 0

y{zzzzzz �; z Ï H-¥, 0L01.13.26.0014.01

cos-1 ikjjjjj �!!!z

��!!!!!!!!!!!z + 1 + 1

y{zzzzz �1

��2 �!!!!!!!2 Π

G3,32,2

ikjjjjjjz


0, 1��2 , 0

y{zzzzzz �; z Ï H-¥, 0LClassical cases involving unit step Θ ÿ

01.13.26.0015.01

ΘH1 - z¤L cos-1 I�!!!z M ��!!!!

Π��

2G2,2

2,0 ikjjjjjzÄÄÄÄÄÄÄÄÄÄÄ 1, 1

0, 1��2

y{zzzzz �; z Ï H-1, 0L01.13.26.0016.01

ΘH z¤ - 1L cos-1 ikjjjj 1

��!!!z y{zzzz ��!!!!

Π��

2G2,2

0,2 ikjjjjjzÄÄÄÄÄÄÄÄÄÄÄ 1��2 , 1

0, 0

y{zzzzz


Generalized cases for the direct function itselfÿ

01.13.26.0017.01

cos-1 HzL �1

��2 z �!!!!

Π G2,2

1,2ikjjjjjj�!!!!!!!!!

-z2 ,1��2


1, 1��2

y{zzzzzz +Π��2

01.13.26.0018.01


+ä

��2 �!!!!

ΠG2,2

1,2 ikjjjjjä z,1��2

ÄÄÄÄÄÄÄÄÄÄÄ 1, 11��2 , 0

y{zzzzzGeneralized cases involving algebraic functions in the argumentsÿ

01.13.26.0019.01

cos-1 I�!!!!!!!!!!!!!z2 + 1 - zM �

1��2 �!!!!!!!2 Π

G3,32,2

ikjjjjjjz,1��2

ÄÄÄÄÄÄÄÄÄÄÄÄÄ 1��2 , 1, 11��4 , 3��4 , 0

y{zzzzzz �; z Ï H-¥, 0L01.13.26.0020.01

cos-1 ikjjjjj �!!!!!!!!!!!!!

z2 + 1 - 1��

z

y{zzzzz �1

��2 �!!!!!!!2 Π

G3,32,2

ikjjjjjjz,1��2


0, 1��2 , 0

y{zzzzzz01.13.26.0021.01

cos-1 ikjjjj 1

��!!!!!!!!!!!!!z2 + 1 + z

y{zzzz �1

��2 �!!!!!!!2 Π

G3,32,2

ikjjjjjjz,1��2

ÄÄÄÄÄÄÄÄÄÄÄÄÄ 1��2 , 1, 11��4 , 3��4 , 0

y{zzzzzz �; z Ï H-¥, 0L01.13.26.0022.01

cos-1 ikjjjj z

��!!!!!!!!!!!!!z2 + 1 + 1

y{zzzz �1

��2 �!!!!!!!2 Π

G3,32,2

ikjjjjjjz,1��2


0, 1��2 , 0

y{zzzzzzGeneralized cases involving unit step Θ ÿ

01.13.26.0023.01

ΘH1 - z¤L cos-1 HzL �1��2

�!!!!Π G2,2

2,0 ikjjjjjz,1��2

ÄÄÄÄÄÄÄÄÄÄÄ 1, 1

0, 1��2

y{zzzzz01.13.26.0024.01

ΘH z¤ - 1L cos-1 ikjj 1��z

y{zz �1��2

�!!!!Π G2,2

0,2 ikjjjjjz,1��2

ÄÄÄÄÄÄÄÄÄÄÄ 1��2 , 1

0, 0

y{zzzzzThrough other functionsÿ

Involving inverse Jacobi functionsÿ

01.13.26.0025.01

cos-1 HzL � cd-1 Hz È 0L01.13.26.0026.01

cos-1 HzL � cn-1 Hz È 0L01.13.26.0027.01

cos-1 HzL � dc-1 ikjj 1��z

ÄÄÄÄÄÄÄÄÄ 0y{zz01.13.26.0028.01


- ds-1 ikjj 1��z

ÄÄÄÄÄÄÄÄÄ 0y{zz


01.13.26.0029.01

cos-1 HzL � nc-1 ikjj 1��z

ÄÄÄÄÄÄÄÄÄ 0y{zz01.13.26.0030.01


- ns-1 ikjj 1��z

ÄÄÄÄÄÄÄÄÄ 0y{zz01.13.26.0031.01


- sd-1 Hz È 0L01.13.26.0032.01


- sn-1 Hz È 0LInvolving some hypergeometric-type functionsÿ

01.13.26.0033.01


-�!!!!!

z2��

2 z Bz2

ikjj 1��2

,1��2

y{zz01.13.26.0034.01

cos-1 HzL �1

��2 z

ikjjΠ Iz -�!!!!!

z2 M +�!!!!!

z2 B1-z2ikjj 1

��2

,1��2

y{zzy{zzRepresentations through equivalent functionsÿ

With inverse functionÿ

01.13.27.0001.01

cos-1 HcosHzLL � z �; 0 < ReHzL < Π Þ HReHzL � 0 ß ImHzL ³ 0L Þ HReHzL � Π ß ImHzL £ 0L01.13.27.0002.01

cos-1 HcosHzLL � -z �; -Π < ReHzL < 0 Þ HReHzL � 0 ß ImHzL £ 0L Þ HReHzL � -Π ß ImHzL ³ 0L01.13.27.0003.01

cos-1 HcosHzLL �Π��2

H1 + H-1Lk L + H-1Lk Hz - Π Hk + 1LL �;Hk Π < ReHzL < Hk + 1L Π Þ HReHzL � k Π ß ImHzL ³ 0L Þ HReHzL � Hk + 1L Π ß ImHzL £ 0LL ß k Î Ù

01.13.27.0004.01

cos-1 HcosHzLL �Π��2

I1 - H-1Le- ReHzL��Π u M + H-1Le- ReHzL��Π u II1 + H-1Le ReHzL��Π u+e- ReHzL��Π u M ΘHImHzLL - 1M ikjjz + Π f-ReHzL��

Πvy{zz

01.13.27.0005.01

cosHcos-1 HzLL � z

With related functionsÿ

Involving logÿ

01.13.27.0006.01


+ ä logIä z +�!!!!!!!!!!!!!

1 - z2 M01.13.27.0007.01

cos-1 HzL � -ä logIz +�!!!!!!!!!!!!!

z2 - 1 M �; ReHzL ImHzL > 0 Þ ReHzL � 0 Þ -1 < z < 1


01.13.27.0008.01

cos-1 HzL ��!!!!!!!!!!!1 - z��!!!!!!!!!!!z - 1

logIz +�!!!!!!!!!!!z - 1 �!!!!!!!!!!!z + 1 M

01.13.27.0009.01

cos-1 HzL � ä IlogH2L - 2 logI�!!!!!!!!!!!z + 1 + ä�!!!!!!!!!!!1 - z MM

01.13.27.0010.01

cos-1 HzL � -2 ä logikjjjjjjj$%%%%%%%%%%%%%z + 1

��2

+ ä $%%%%%%%%%%%%%1 - z��

2

y{zzzzzzzInvolving sin-1 ÿ

01.13.27.0011.01


- sin-1 HzL01.13.27.0012.01

cos-1 I�!!!!!!!!!!!1 - z M � sin-1 I�!!!z M01.13.27.0013.01

cos-1ikjjjjjjj$%%%%%%%%%%%%%1 - z

��2


sin-1 HzL +Π��4

01.13.27.0014.01

cos-1 H-zL � sin-1 HzL +Π��2

01.13.27.0015.01

cos-1 H1 - 2 z2 L �2

�!!!!!z2

��z

sin-1 HzL01.13.27.0016.01

cos-1 I�!!!!!!!!!!!!!1 - z2 M �

�!!!!!z2

��z

sin-1 HzL01.13.27.0017.01

cos-1ikjjjjjjj$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%1

��2

I1 -�!!!!!!!!!!!!!

1 - z2 M y{zzzzzzz �Π��2

-�!!!!!

z2��

2 z sin-1 HzL

Involving tan-1 ÿ

01.13.27.0018.01

cos-1 HzL � 2 tan-1ikjjjjjjj$%%%%%%%%%%%%%1 - z

��z + 1

y{zzzzzzz �; z Ï H-¥, -1L01.13.27.0019.01

cos-1 ikjj 2 z��z2 + 1

y{zz �Π��2

- 2 tan-1 HzL �; z¤ < 1

01.13.27.0020.01

cos-1 ikjjj 1 - z2

��1 + z2

y{zzz �2

�!!!!!z2

��z

tan-1 HzL �; ä z Ï H-¥, -1L ß ä z Ï H1, ¥L01.13.27.0021.01

cos-1 ikjjjj 1��!!!!!!!!!!!z + 1

y{zzzz � tan-1 I�!!!z M


01.13.27.0022.01

cos-1 ikjjjj 1��!!!!!!!!!!!!!

z2 + 1

y{zzzz ��!!!!!

z2��

z tan-1 HzL

01.13.27.0023.01

cos-1 ikjjjj z��!!!!!!!!!!!!!

z2 + 1

y{zzzz �Π��2

- tan-1 HzL �; ä z Ï H-¥, -1L ß ä z Ï H1, ¥L01.13.27.0024.01

cos-1 ikjjjj r zc

��!!!!!!!!!!!!!!!!!!!!r2 z2 c + 1

y{zzzz �Π��2

ikjjjjjjj$%%%%%%%%%%%%%%%%%%%%%1��ä r zc + 1

�!!!!!!!!!!!!!!!!!!ä r zc + 1 - $%%%%%%%%%%%%%%%%%%%%%1

��1 - ä r zc

�!!!!!!!!!!!!!!!!!!1 - ä r zc + 1y{zzzzzzz - tan-1 Hr zc L $%%%%%%%%%%%%%%%%%%%%%%%1

��r2 z2 c + 1

�!!!!!!!!!!!!!!!!!!!!r2 z2 c + 1

01.13.27.0025.01

cos-1ikjjjjj �!!!z

��!!!!!!!!!!!a + z


-�!!!!!!!!!!!a + z��

2 �!!!z $%%%%%%%%%%%%%z��a + z

ikjjjjjjΠ - $%%%%%%%%%%%%%a��a + z

$%%%%%%%%%%%%%%z��a

+ 1ikjjjjjjΠ - 2 tan-1

ikjjjjjj$%%%%%%z��a

y{zzzzzzy{zzzzzzy{zzzzzz01.13.27.0026.01


��!!!!!!!!!!!z + 1

y{zzzzz �1��2

$%%%%%%%%%%%%%1��z + 1

�!!!!!!!!!!!z + 1 IΠ - 2 tan-1 I�!!!z MM01.13.27.0027.01

cos-1ikjjjjjj$%%%%%%%%%%%%%z

��z + 1

y{zzzzzz �1��2

$%%%%%%%%%%%%%1��z + 1

�!!!!!!!!!!!z + 1 IΠ - 2 tan-1 I�!!!z MM01.13.27.0028.01


��!!!!!!!!!!!z + 1


+�!!!z

��"#########z��z+1�!!!!!!!!!!!z + 1

ikjjjjjjjj Π

��2

ikjjjjjjj1 -�!!!z

��!!!!!!!!!!!z + 1 $%%%%%%%%%%%%%z + 1

��z

y{zzzzzzz +cot-1 I�!!!z M

��"#####1��z�!!!!!!!!!!!z + 1

$%%%%%%%%%%%%%z + 1��

z-

Π��2

y{zzzzzzzzInvolving cot-1 ÿ

01.13.27.0029.01

cos-1 ikjj 2 z��z2 + 1

y{zz �Π��2

- 2 cot-1 HzL �; z¤ > 1

01.13.27.0030.01

cos-1 ikjjjj z��!!!!!!!!!!!!!

z2 + 1

y{zzzz �Π��2

-z

��!!!!!!!!!!!!!z2 + 1

$%%%%%%%%%%%%%%%1 +1

��z2

ikjjjjjjj Π��2

- $%%%%%%%%1��z2

z cot-1 HzLy{zzzzzzz01.13.27.0031.01

cos-1 ikjjjj 1��!!!!!!!!!!!!!!!!a zc + 1

y{zzzz � -�!!!!a zc�2��!!!!!!!!!a zc

cot-1 I�!!!!a zc�2 M +Π ä��2

ikjjjjjjjj "#################################-ä

�!!!!a �!!!!!zc - 1��"#############################

ä�!!!!a �!!!!!zc + 1

+"#############################

ä�!!!!a �!!!!!zc - 1

��"#############################1 - ä

�!!!!a �!!!!!zc

y{zzzzzzzz +Π��2

Involving csc-1 ÿ

01.13.27.0032.01

cos-1ikjjjjjjj$%%%%%%%%%%%%%z - 1

��z

y{zzzzzzz ��!!!z $%%%%%%1

��z

csc-1 I�!!!z M01.13.27.0033.01

cos-1ikjjjjj �!!!!!!!!!!!z - 1

��!!!z y{zzzzz ��!!!z $%%%%%%1

��z

csc-1 I�!!!z M


01.13.27.0034.01

cos-1ikjjjjjjj$%%%%%%%%%%%%%z - 1

��2 z


csc-1 HzL +Π��4

01.13.27.0035.01

cos-1ikjjjjj �!!!!!!!!!!!z - 1

��!!!!!!2 z

y{zzzzz �1��2

csc-1 HzL +Π��4

01.13.27.0036.01

cos-1ikjjjjj �!!!!!!!!!!!!!

z2 - 1��

z


ikjjjjj1 -�!!!!!

z2��

z

y{zzzzz + csc-1 HzL �; ReHzL ¹ 0

01.13.27.0037.01

cos-1ikjjjjj �!!!!!!!!!!!!!

z2 - 1��

z


ikjjjjj1 -�!!!!!

z2��

z

y{zzzzz +�!!!!!

z2 $%%%%%%%%1��z2

csc-1 HzL01.13.27.0038.01

cos-1ikjjjjjjjr z-c $%%%%%%%%%%%%%%%%%%z2 c

��r2

- 1y{zzzzzzz �

Π��2

-�!!!!!!!!!!!!!!!

r2 z-2 c $%%%%%%%%%%z2 c��r2

ikjjjjj Π zc �!!!!!!!!!!!!!!!r2 z-2 c

��2 r

- csc-1 J zc

��r

Ny{zzzzz01.13.27.0039.01

cos-1ikjjjjj �!!!!!!!!!!!z - a

��!!!z y{zzzzz �Π��2

-�!!!!!!!!!!!z - a

��2 "#############1 - a��z

�!!!z ikjjjjjjΠ - 2 $%%%%%%a��z

$%%%%%%z��a

csc-1ikjjjjjj$%%%%%%z

��a

y{zzzzzzy{zzzzzz01.13.27.0040.01

cos-1ikjjjjjjj$%%%%%%%%%%%%%z - 1

��z

y{zzzzzzz �Π��2

-�!!!z

��!!!!!!!!!!!z - 1 $%%%%%%%%%%%%%z - 1

��z

ikjjjjjjj Π��2

- $%%%%%%1��z

�!!!z csc-1 I�!!!z My{zzzzzzzInvolving sec-1 ÿ

01.13.27.0041.01

cos-1 HzL � sec-1 ikjj 1��z

y{zz01.13.27.0042.01

cos-1ikjjjjjjj$%%%%%%%%%%%%%z + 1

��2 z


sec-1 HzL01.13.27.0043.01

cos-1ikjjjjj �!!!!!!!!!!!z + 1

��!!!!!!2 z

y{zzzzz �1��2

sec-1 HzL �; z Ï H-1, 0LInvolving sinh-1 ÿ

01.13.27.0044.01

cos-1 HzL � ä sinh-1 Hä zL +Π��2

01.13.27.0045.01

cos-1 I�!!!!!!-z M �

Π��2

-�!!!!!!!!!

-z2��

zsinh-1 I�!!!z M

01.13.27.0046.01

cos-1 I�!!!!!!!!!!!z + 1 M ��!!!!!!!!!

-z2��

z sinh-1 I�!!!z M


01.13.27.0047.01

cos-1 I�!!!!!!!!!!!!!1 + z2 M �

�!!!!!!!!!-z2

��z

sinh-1 HzL01.13.27.0048.01

cos-1 I�!!!!!!!!!!!!!!!!a zc + 1 M � -�!!!!a zc�2��!!!!!!!!!!!!

-a zc sinh-1 I�!!!!a zc�2 M

Involving cosh-1 ÿ

01.13.27.0049.01

cos-1 HzL ��!!!!!!!!!!!1 - z��!!!!!!!!!!!z - 1

cosh-1 HzL01.13.27.0050.01

cos-1 I�!!!z M ��!!!!!!!!!!!1 - z��!!!!!!!!!!!z - 1

cosh-1 I�!!!z M01.13.27.0051.01

cos-1 HzL � -ä cosh-1 HzL �; ImHzL > 0

01.13.27.0052.01

cos-1 HzL � ä cosh-1 HzL �; ImHzL < 0

Involving tanh-1 ÿ

01.13.27.0053.01

cos-1 ikjjj z2 + 1��1 - z2

y{zzz �2

�!!!!!!!!!-z2

��z

tanh-1 HzL �; z Ï H-¥, -1L ß z Ï H1, ¥L01.13.27.0054.01

cos-1 ikjjjj 1��!!!!!!!!!!!1 - z

y{zzzz ��!!!!!!!!!

-z2��

z tanh-1 I�!!!z M

01.13.27.0055.01

cos-1 ikjjjj z��!!!!!!!!!!!!!

z2 - 1

y{zzzz ��!!!!!!!!!!!!!

z2 - 1 $%%%%%%%%%%%%%%%1��1 - z2

tanh-1 HzL +Π��2

�; z Ï H-¥, -1L ß z Ï H1, ¥L01.13.27.0056.01

cos-1 ikjjjj z��!!!!!!!!!!!!!

z2 - 1

y{zzzz �Π��2

ikjjjjjjj-$%%%%%%%%%%%%%1

��z + 1

�!!!!!!!!!!!z + 1 + $%%%%%%%%%%%%%1��1 - z

�!!!!!!!!!!!1 - z + 1y{zzzzzzz +

�!!!!!!!!!!!!!z2 - 1 $%%%%%%%%%%%%%%%1

��1 - z2

tanh-1 HzL01.13.27.0057.01

cos-1 ikjjjj r zc

��!!!!!!!!!!!!!!!!!!!!r2 z2 c - 1

y{zzzz �Π��2

ikjjjjjjj-$%%%%%%%%%%%%%%%%%%1��r zc + 1

�!!!!!!!!!!!!!!!!r zc + 1 + $%%%%%%%%%%%%%%%%%%1��1 - r zc

�!!!!!!!!!!!!!!!!1 - r zc + 1y{zzzzzzz +

�!!!!!!!!!!!!!!!!!!!!r2 z2 c - 1 $%%%%%%%%%%%%%%%%%%%%%%%1

��1 - r2 z2 c

tanh-1 Hr zc L01.13.27.0058.01


��!!!!!!!!!!!z - a


-�!!!!!!!!!!!z - a

��2 �!!!z $%%%%%%%%%%%%%z

��z - a

ikjjjjjjj 1

��z

$%%%%%%%%%%%%%%1 -z��a

$%%%%%%%%%%%%%a��a - z

ikjjjjjjj2 a $%%%%%%%%%%%%-

z2��a2

tanh-1ikjjjjjj$%%%%%%z

��a

y{zzzzzz - Π zy{zzzzzzz + Π

y{zzzzzzz01.13.27.0059.01

cos-1ikjjjjjj$%%%%%%%%%%%%%z

��z - 1

y{zzzzzz � $%%%%%%%%%%%%%1��1 - z

�!!!!!!!!!!!1 - zikjjjjj Π

��2

-�!!!!!!

-z��!!!z tanh-1 I�!!!z My{zzzzz


Involving coth-1 ÿ

01.13.27.0060.01

cos-1 ikjjjj z��!!!!!!!!!!!!!

z2 - 1

y{zzzz �Π��2

ikjjjjj1 -�!!!!!

z2��

z

y{zzzzz +�!!!!!!!!!!!!!

z2 - 1 $%%%%%%%%%%%%%%%1��1 - z2

coth-1 HzL01.13.27.0061.01

cos-1 ikjjjj z��!!!!!!!!!!!!!

z2 - 1

y{zzzz �Π��2

-z

��!!!!!!!!!!!!!z2 - 1

$%%%%%%%%%%%%%%%1 -1

��z2

ikjjjjjjj Π��2

- $%%%%%%%%%%%-1

��z2

z coth-1 HzLy{zzzzzzz01.13.27.0062.01


��!!!!!!!!!!!z - 1

y{zzzzz ��!!!z

��!!!!!!!!!!!z - 1 "#########z��z-1

ikjjjjjjj Π

��2

ikjjjjjjj1 - $%%%%%%%%%%%%%z - 1��

z$%%%%%%%%%%%%%z

��z - 1

y{zzzzzzz +�!!!!!!!!!

-z2��!!!!!!!!!!!1 - z

$%%%%%%%%%%%%%z - 1��

z$%%%%%%%%%%-

1��z

coth-1 I�!!!z M -Π��2

y{zzzzzzz +Π��2

Involving csch-1 ÿ

01.13.27.0063.01

cos-1ikjjjjjjj$%%%%%%%%%%%%%z + 1

��z

y{zzzzzzz ��!!!z $%%%%%%%%%%-

1��z

csch-1 I�!!!z M01.13.27.0064.01

cos-1ikjjjjj �!!!!!!!!!!!z + 1


ikjjjjjjj1 - $%%%%%%%%%%%%%%1 +1��z

$%%%%%%%%%%%%%z��z + 1

y{zzzzzzz + $%%%%%%%%%%%%%z��z + 1

$%%%%%%%%%%%%%%1 +1��z

$%%%%%%%%%%-1��z

�!!!z csch-1 I�!!!z M01.13.27.0065.01

cos-1ikjjjjj �!!!!!!!!!!!!!

z2 + 1��

z

y{zzzzz � $%%%%%%%%%%%-1

��z2

z csch-1 HzL �; ReHzL > 0

01.13.27.0066.01

cos-1ikjjjjj �!!!!!!!!!!!!!

z2 + 1��

z


ikjjjjj1 -�!!!!!

z2��

z

y{zzzzz -�!!!!!!!!!

-z4 csch-1 HzL��

z2�; ReHzL ImHzL ¹ 0

01.13.27.0067.01

cos-1ikjjjjj �!!!!!!!!!!!!!

z2 + 1��

z


ikjjjjjjj1 -z

��!!!!!!!!!!!!!z2 + 1

$%%%%%%%%%%%%%%%z2 + 1��

z2

y{zzzzzzz -�!!!!!!!!!!!!!

z2 + 1 csch-1 HzL��

z2 "#############-z2 -1��z4

01.13.27.0068.01

cos-1ikjjjjjjjr z-c $%%%%%%%%%%%%%%%%%%z2 c

��r2

+ 1y{zzzzzzz �

Π��2

-r

��2 z2 c �!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

-r2 z-4 c Hz2 c + r2 L $%%%%%%%%%%%%%%%%%%z2 c��r2

+ 1ikjjjjjjjΠ $%%%%%%%%%%%%%-

r2��z2 c

zc + 2 r csch-1 J zc

��r

Ny{zzzzzzz01.13.27.0069.01

cos-1ikjjjjj �!!!!!!!!!!!a + z


-�!!!!!!!!!!!a + z

��"#############a��z + 1 �!!!z ikjjjjjj Π

��2

- $%%%%%%%%%%-a��z

$%%%%%%z��a

csch-1ikjjjjjj$%%%%%%z

��a

y{zzzzzzy{zzzzzz01.13.27.0070.01

cos-1ikjjjjj �!!!!!!!!!!!z + 1

��!!!z y{zzzzz ��!!!!!!!!!!!z + 1

��!!!z "#########z+1��z

ikjjjjjjj$%%%%%%%%%%-

1��z

�!!!z csch-1 I�!!!z M -Π��2

y{zzzzzzz +Π��2

01.13.27.0071.01

cos-1ikjjjjj �!!!!!!!!!!!z + 1


-�!!!!!!

-z �!!!!!!!!!!!z + 1��!!!!!!!!!!!!!!

-z - 1 �!!!z ikjjjjjjj Π

��2

- $%%%%%%%%%%-1��z

�!!!z csch-1 I�!!!z My{zzzzzzz


Involving sech-1 ÿ

01.13.27.0072.01

cos-1 ikjjjj 1��!!!z y{zzzz �

�!!!!!!!!!!!z - 1��!!!!!!!!!!!1 - z

sech-1 I�!!!z MInequalitiesÿ

01.13.29.0001.01

cos-1 HxL ³ 0 �; x¤ £ 1 ß x Î Ñ

01.13.29.0002.01

cos-1 HxL £ Π �; x¤ £ 1 ß x Î Ñ

Zerosÿ

01.13.30.0001.01

cos-1 HzL � 0 �; z � 1

Historyÿ

| J. Herschel (1813) introduced the notation cos-1

The function cos-1 is often encountered in mathematics and the natural sciences.


Copyrightÿ

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 http://functions.wolfram.com/Notations/.

Please cite this document by referring to the functions.wolfram.com page from which it was downloaded, for example:

http://functions.wolfram.com/Constants/E/

To refer to a particular formula, cite functions.wolfram.com followed by the citation number.e.g.: http://functions.wolfram.com/01.03.03.0001.01

This document is currently in a preliminary form. If you have comments or suggestions, please email [email protected].

© 2001, Wolfram Research, Inc.


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