Arg
Notations
Traditional name
Argument
Traditional notation
argHzL
Mathematica StandardForm notation
Arg@zD
Primary definition
12.02.02.0001.01
argHzL Š -ä log
z
z¤
ArgHzL is the argument of z, such that z Š z¤ ãä ArgHzL . The argument of a complex number z is the phase angle (in
radians) that the line from 0 to z makes with the positive real axis.
Specific values
Specialized values
argHxL Š 0 ; x Î R ß x > 0
12.02.03.0001.01
argHxL Š Π ; x Î R ß x < 0
12.02.03.0020.01
12.02.03.0002.01
argHä xL Š
Π
2
; x Î R ß x > 0
12.02.03.0021.01
argHä xL Š -
Π
2
; x Î R ß x < 0
argHx + ä yL Š tan-1 Hx, yL ; x Î R ì y Î R
12.02.03.0003.01
Values at fixed points
http://functions.wolfram.com
argH0L Î H-Π, ΠD
12.02.03.0004.01
ArgH0L is not a uniquely defined number. Depending on the argument of z, the limit limÈzÈ®0 ArgHzL can take any
value in the interval H-Π, ΠL.
12.02.03.0005.01
argH1L Š 0
12.02.03.0006.01
argH-1L Š Π
12.02.03.0007.01
argHäL Š
Π
2
12.02.03.0008.01
argH-äL Š -
Π
2
12.02.03.0022.01
argH1 + äL Š
Π
4
12.02.03.0023.01
argH-1 + äL Š
3Π
4
12.02.03.0024.01
argH-1 - äL Š -
3Π
4
12.02.03.0025.01
argH1 - äL Š -
Π
4
12.02.03.0026.01
argI 3 + äN Š
Π
6
12.02.03.0027.01
3 NŠ
argI1 + ä
Π
3
12.02.03.0028.01
argI-1 + ä
3 NŠ
2Π
3
12.02.03.0029.01
argI- 3 + äN Š
5Π
6
12.02.03.0030.01
argI- 3 - äN Š -
5Π
6
2
http://functions.wolfram.com
3
12.02.03.0031.01
3 NŠ-
argI-1 - ä
2Π
3
12.02.03.0032.01
3 NŠ-
argI1 - ä
Π
3
12.02.03.0033.01
argI 3 - äN Š -
Π
6
12.02.03.0009.01
argH2L Š 0
12.02.03.0010.01
argH-2L Š Π
12.02.03.0011.01
argHΠL Š 0
12.02.03.0012.01
argH3 äL Š
Π
2
12.02.03.0013.01
argH-2 äL Š -
Π
2
12.02.03.0014.01
argH2 + äL Š tan-1
1
2
Values at infinities
12.02.03.0015.01
argH¥L Š 0
12.02.03.0016.01
argH-¥L Š Π
12.02.03.0017.01
argHä ¥L Š
Π
2
12.02.03.0018.01
argH-ä ¥L Š -
Π
2
Ž L Î H-Π, ΠD
argH¥
12.02.03.0019.01
General characteristics
Domain and analyticity
ArgHzL is a nonanalytical function; it is a real-analytic function of the complex variable z for z ¹ 0.
http://functions.wolfram.com
4
12.02.04.0001.01
z™argHzL › C™R
Symmetries and periodicities
Parity
ArgHzL is an odd function for almost all z.
argH-zL Š -argHzL ; z Ï H-¥, 0L
12.02.04.0002.01
12.02.04.0003.01
argH-zL Š argHzL -
-z
äΠ
z
Mirror symmetry
argHz L Š -argHzL ; z Ï H-¥, 0L
12.02.04.0004.01
Periodicity
No periodicity
Homogeneity
argHa zL Š argHzL ; a Î R ß a > 0
12.02.04.0005.01
Sets of discontinuity
The function ArgHzL is a single-valued, continuous function on the z-plane cut along the interval H-¥, 0L, where it
is continuous from above.
DSz HargHzLL Š 88H-¥, 0L, -ä<<
12.02.04.0006.01
lim argHx + ä ΕL Š argHxL Š Π ; x Î R ß x < 0
12.02.04.0007.01
Ε®+0
lim argHx - ä ΕL Š -Π ; x Î R ß x < 0
12.02.04.0008.01
Ε®+0
Transformations
Transformations and argument simplifications
Argument involving complex characteristics
12.02.16.0032.01
argH z¤L Š 0
http://functions.wolfram.com
5
12.02.16.0033.01
z
arg
z¤
Š argHzL
12.02.16.0034.01
argHsgnHzLL Š argHzL
argHz L Š -argHzL ; argHzL ¹ Π
12.02.16.0006.01
12.02.16.0035.01
argHzL + Π
argHz L Š 2 Π
- argHzL
2Π
Argument involving basic arithmetic operations
argH-zL Š -argHzL ; z Ï H-¥, 0L
12.02.16.0001.01
12.02.16.0002.01
-z
argH-zL Š argHzL -
äΠ
z
12.02.16.0036.01
argH-zL Š argHzL + Π 2 -
argHzL
+1
2Π
12.02.16.0037.01
argHä zL Š argHzL +
; argHzL £
Π
2
Π
2
12.02.16.0038.01
argHä zL Š argHzL -
3Π
2
; argHzL >
Π
2
H-1L34 Π
12.02.16.0004.01
argHä zL Š argHzL -
Π
2
-
äz
z
12.02.16.0039.01
argHä zL Š argHzL + 2 Π
1
4
-
argHzL
2Π
+
Π
2
12.02.16.0040.01
argH-ä zL Š argHzL -
Π
2
; argHzL > -
12.02.16.0041.01
argH-ä zL Š argHzL +
3Π
2
Π
2
; argHzL £ -
Π
2
12.02.16.0005.01
argH-ä zL Š argHzL +
Π
2
4
-
-1 Π
-ä z
z
http://functions.wolfram.com
6
12.02.16.0042.01
3
argH-ä zL Š argHzL + 2 Π
4
-
argHzL
-
2Π
Π
2
12.02.16.0007.01
Š -argHzL ; z Ï H-¥, 0L
1
arg
z
12.02.16.0008.01
Š -argHzL ; argHzL ¹ Π
1
arg
z
12.02.16.0043.01
Š 2 Π - argHzL ; argHzL Š Π
1
arg
z
12.02.16.0009.01
1
arg
1
Š- z
z
argHzL
z
12.02.16.0044.01
1
arg
Š 2Π
z
argHzL + Π
- argHzL
2Π
12.02.16.0045.01
arg -
1
z
Š Π - argHzL ; ImHzL ³ 0
12.02.16.0046.01
arg -
1
z
Š -argHzL - Π ; ImHzL < 0
12.02.16.0047.01
arg -
1
z
Š -argHzL - Π ä
-
1
z
z
12.02.16.0048.01
arg -
1
z
Š -argHzL + 2 Π
argHzL
2Π
+Π
12.02.16.0049.01
ä
arg
Š
z
Π
2
- argHzL ; argHzL ³ -
12.02.16.0050.01
ä
arg
Š -argHzL -
z
3Π
2
Π
2
; argHzL < -
Π
2
12.02.16.0051.01
ä
arg
Š -argHzL + 2 Π
z
argHzL
2Π
+
1
4
+
12.02.16.0052.01
arg -
ä
z
Š-
Πä
2
- argHzL ; argHzL <
Π
2
Π
2
http://functions.wolfram.com
7
12.02.16.0053.01
arg -
ä
z
Š
3Π
2
- argHzL ; argHzL ³
Π
2
12.02.16.0054.01
arg -
ä
z
Š -argHzL + 2 Π
argHzL
2Π
+
3
4
-
Π
2
Addition formulas
argHx + ä yL Š tan-1 Hx, yL ; ImHxL Š 0 ì ImHyL Š 0
12.02.16.0010.01
Multiple arguments
For products
argHa zL Š argHzL ; a Î R ß a > 0
12.02.16.0011.01
argHz1 z2 L Š argHz1 L + argHz2 L ; -Π < argHz1 L + argHz2 L £ Π
12.02.16.0014.01
argIz - z M Š argH1 - zL + argHzL
12.02.16.0055.01
2
argI-z - z2 M Š argH1 + zL + argH-zL
12.02.16.0056.01
argHz1 z2 L Š argHz1 L + argHz2 L ; argHz1 L £ 0 ì -argHz1 L - Π < argHz2 L ê argHz1 L ³ 0 ì argHz2 L £ Π - argHz1 L
12.02.16.0057.01
argHz1 z2 L Š argHz1 L + argHz2 L - 2 Π ; argHz1 L ³ 0 ì argHz2 L > Π - argHz1 L
12.02.16.0058.01
argHz1 z2 L Š argHz1 L + argHz2 L + 2 Π ; argHz1 L £ 0 ì argHz2 L £ -argHz1 L - Π
12.02.16.0059.01
12.02.16.0015.01
argHz1 z2 L Š argHz1 L + argHz2 L + 2 Π
12.02.16.0060.01
arg ä zk Š â argHzk L + 2 Π
n
n
k=1
k=1
12.02.16.0061.01
arg
z+1
Š argHzL - argHz + 1L
12.02.16.0062.01
z
arg
z-1
2Π
Π - Únk=1 argHzk L
For quotients
z
Π - argHz1 L - argHz2 L
Š argH-zL - argH1 - zL
2Π
; n Î N+
http://functions.wolfram.com
8
12.02.16.0016.01
arg
z1
z2
Š argHz1 L - argHz2 L ; -Π < argHz1 L - argHz2 L £ Π
12.02.16.0063.01
arg
z1
z2
Š argHz1 L - argHz2 L ; argHz1 L £ 0 ì argHz2 L < argHz1 L + Π ê argHz1 L > 0 ì argHz2 L ³ argHz1 L - Π
12.02.16.0064.01
arg
z1
z2
Š argHz1 L - argHz2 L - 2 Π ; argHz1 L ³ 0 ì argHz2 L < argHz1 L - Π
12.02.16.0065.01
arg
z1
z2
Š argHz1 L - argHz2 L + 2 Π ; argHz1 L £ 0 ì argHz2 L ³ argHz1 L + Π
12.02.16.0017.01
arg
z1
z2
Š argHz1 L - argHz2 L + 2 Π
Π - argHz1 L + argHz2 L
2Π
Power of arguments
12.02.16.0066.01
argI z N Š
argHzL
2
12.02.16.0067.01
arg
Š argHzL ; ReHzL > 0 Þ ReHzL Š 0 ß ImHzL > 0
z2
12.02.16.0068.01
Πä
arg
z2 - z
Š argHzL -
z2
-z2
2
12.02.16.0069.01
argIz1n M Š
argHzL
n
; n Î Z ß n ¹ 0 ß n ¹ -1
argIz2 M Š 2 argHzL ; ReHzL > 0 Þ ReHzL Š 0 ß ImHzL > 0
12.02.16.0070.01
12.02.16.0071.01
argIz2 M Š 2 argHzL + 2 Π ; -Π < argHzL £ 12.02.16.0072.01
argIz2 M Š 2 argHzL - 2 Π ;
Π
2
< argHzL £ Π
12.02.16.0073.01
argIz2 M Š 2 argHzL + 2 Π
1
2
-
argHzL
Π
Π
2
http://functions.wolfram.com
9
12.02.16.0074.01
argIz M Š 2 argHzL -
Πä
z2 - z
2
-z2
argHx L Š tan-1 HcosHImHaL logHxLL, sinHImHaL logHxLLL ; x Î R ß x > 0
12.02.16.0018.01
a
argHza L Š a argHzL ; a Î R ì -Π < a argHzL £ Π
12.02.16.0019.01
argHza L Š a argHzL + 2 Π k ; a Î R ì -Π - 2 Π k < a argHzL £ Π - 2 Π k ì k Î Z
12.02.16.0075.01
argHza L Š ImHa logHzLL ; -Π < ImHa logHzLL £ Π
12.02.16.0076.01
argHza L Š ImHa logHzLL + 2 Π k ; -2 Π k - Π < ImHa logHzLL £ Π - 2 Π k ì k Î Z
12.02.16.0077.01
argHz L Š argIãä a argHzL M ; a Î R
12.02.16.0020.01
a
argHz L Š tan-1 IcosIa tan-1 HReHzL, ImHzLLM, sinIa tan-1 HReHzL, ImHzLLMM ; a Î R
12.02.16.0021.01
a
12.02.16.0022.01
argHx L Š ImHaL logHxL ; a
Π
logHxL
12.02.16.0024.01
argHza L Š a argHzL + 2 Π
Π
< ImHaL £
Π - a argHzL
2Π
logHxL
íx ÎRíx >0
; ImHaL Š 0
12.02.16.0078.01
argHza L ‡ a argHzL + ImHaL logHzL + 2 Π
Π - ImHa logHzLL
2Π
12.02.16.0079.01
argHza L ‡ a argHzL + 2 Π
Π - ImHa logHzLL
2Π
- ä a logH z¤L + ä ReHa logHzLL
12.02.16.0025.01
argHza L Š 2 Π
Π - ImHa logHzLL
2Π
+ ImHa logHzLL
12.02.16.0080.01
argHza L ‡ ImHaL logH z¤L + argHzL ReHaL + 2 Π
Π - ImHa logHzLL
2Π
12.02.16.0026.01
argHza L Š ReHaL argHzL + Im HaL logH z¤L + 2 Π
Π - ImHaL logH z¤L - argHzL ReHaL
2Π
argHza L Š tan-1 IcosIImHaL logH z¤L + tan-1 HReHzL, ImHzLL ReHaLM, sinIImHaL logH z¤L + tan-1 HReHzL, ImHzLL ReHaLMM
12.02.16.0027.01
http://functions.wolfram.com
10
Exponent of arguments
argIãx+ä y M Š tan-1 HcosHyL, sinHyLL
12.02.16.0081.01
argHãz L Š ImHzL ; -Π < ImHzL £ Π
12.02.16.0082.01
argHãz L Š 2 Π k + ImHzL ; -2 Π k - Π < ImHzL £ Π - 2 Π k ì k Î Z
12.02.16.0083.01
12.02.16.0084.01
argHãz L Š ImHzL + 2 Π
Π - ImHzL
2Π
12.02.16.0085.01
argIãä z M Š ReHzL + 2 Π
Π - ReHzL
2Π
argHãz L Š Π - HΠ - ImHzLL mod H2 ΠL
12.02.16.0086.01
argIãä z M Š Π - HΠ - ReHzLL mod H2 ΠL
12.02.16.0087.01
Some functions of arguments
12.02.16.0088.01
argHc za L Š argHcL + ImHa logHzLL + 2 Π
Π - argHcL - ImHa logHzLL
2Π
12.02.16.0089.01
argHc ãz L Š argHcL + ImHzL + 2 Π
Π - argHcL - ImHzL
2Π
12.02.16.0090.01
argIxa yb M ‡ a argHxL + b argHyL + 2 Π
-a argHxL - b argHyL + Π
2Π
; a Î R ì b Î R
12.02.16.0091.01
argIxa yb zc M ‡ a argHxL + b argHyL + c argHzL + 2 Π
12.02.16.0092.01
arg ä zk k ‡ â ak argHzk L + 2 Π
n
k=1
a
n
k=1
-a argHxL - b argHyL - c argHzL + Π
2Π
Π - Únk=1 ak argHzk L
2Π
; a Î R ì b Î R ì c Î R
; ak Î R ì 1 £ k £ n
12.02.16.0093.01
argIxa yb M ‡ 2 Π
Π - ImHa logHxLL - ImHb logHyLL
2Π
+ ImHa logHxLL + ImHb logHyLL
12.02.16.0094.01
argIxa yb zc M ‡ 2 Π
Π - ImHa logHxLL - ImHb logHyLL - ImHc logHzLL
2Π
+ ImHa logHxLL + ImHb logHyLL + ImHc logHzLL
http://functions.wolfram.com
11
Π - Únk=1 ImHak logHzk LL
12.02.16.0095.01
ak
ä zk
k=1
n
arg
‡ 2Π
2Π
+ â ImHak logHzk LL
n
k=1
Products, sums, and powers of the direct function
Sums of the direct function
argHz1 L + argHz2 L Š argHz1 z2 L ; argHz1 L £ 0 ì -argHz1 L - Π < argHz2 L ê argHz1 L ³ 0 ì argHz2 L £ Π - argHz1 L
12.02.16.0096.01
argHz1 L + argHz2 L Š argHz1 z2 L + 2 Π ; argHz1 L ³ 0 ì argHz2 L > Π - argHz1 L
12.02.16.0097.01
argHz1 L + argHz2 L Š argHz1 z2 L - 2 Π ; argHz1 L £ 0 ì argHz2 L £ -argHz1 L - Π
12.02.16.0098.01
12.02.16.0028.01
argHz1 L + argHz2 L Š argHz1 z2 L - 2 Π
12.02.16.0099.01
â argHzk L Š arg ä zk - 2 Π
n
k=1
n
Π - argHz1 L - argHz2 L
2Π
Π - Únk=1 argHzk L
; n Î N+
2Π
k=1
Differences of the direct function
12.02.16.0100.01
argHz1 L - argHz2 L Š arg
z1
z2
12.02.16.0101.01
argHz1 L - argHz2 L Š arg
z1
z2
12.02.16.0102.01
argHz1 L - argHz2 L Š arg
z1
z2
12.02.16.0103.01
argHz1 L - argHz2 L Š arg
z1
z2
; -Π < argHz1 L - argHz2 L £ Π
; argHz1 L £ 0 ì argHz2 L < argHz1 L + Π ê argHz1 L > 0 ì argHz2 L ³ argHz1 L - Π
+ 2 Π ; argHz1 L ³ 0 ì argHz2 L < argHz1 L - Π
- 2 Π ; argHz1 L £ 0 ì argHz2 L ³ argHz1 L + Π
12.02.16.0104.01
argHz1 L - argHz2 L Š arg
z1
z2
-2Π
argHz2 L - argHz1 L + Π
2Π
Linear combinations of the direct function
12.02.16.0105.01
a argHxL + b argHyL ‡ argIxa yb M - 2 Π
Π - a argHxL - b argHyL
2Π
; a Î R ì b Î R
12.02.16.0106.01
a argHxL + b argHyL + c argHzL ‡ argIxa yb zc M - 2 Π
Π - a argHxL - b argHyL - c argHzL
2Π
; a Î R ì b Î R ì c Î R
http://functions.wolfram.com
12
12.02.16.0107.01
â ak argHzk L ‡ arg
n
k=1
ak
ä zk
k=1
n
-2Π
Π - Únk=1 ak argHzk L
2Π
; ak Î R ì 1 £ k £ n
Related transformations
12.02.16.0029.01
ä argHzL
ã
Š
z
z¤
12.02.16.0030.01
ä argHzL
ã
Š cosHargHzLL + ä sinHargHzLL
12.02.16.0031.01
ãä argHzL Š cos tan-1
ImHzL
ReHzL
+ ä sin tan-1
Complex characteristics
Real part
ReHargHx + ä yLL Š tan-1 Hx, yL
12.02.19.0001.01
12.02.19.0008.01
ReHargHzLL Š argHzL
Imaginary part
12.02.19.0002.01
ImHargHx + ä yLL Š 0
12.02.19.0003.01
ImHargHzLL Š 0
Absolute value
12.02.19.0004.01
argHx + ä yL¤ Š
tan-1 Hx, yL
2
12.02.19.0009.01
argHzL¤ Š
tan-1 HReHzL, ImHzLL
2
Argument
argHargHx + ä yLL Š tan-1 Itan-1 Hx, yL, 0M
12.02.19.0005.01
argHargHzLL Š tan-1 Itan-1 HReHzL, ImHzLL, 0M
12.02.19.0010.01
ImHzL
ReHzL
; -
Π
2
< argHReHzLL £
Π
2
http://functions.wolfram.com
13
Conjugate value
argHx + ä yL Š tan-1 Hx, yL
12.02.19.0006.01
12.02.19.0007.01
argHzL Š argHzL
Signum value
sgnHargHx + ä yLL Š sgnItan-1 Hx, yLM
12.02.19.0011.01
tan-1 Hx, yL
12.02.19.0012.01
sgnHargHx + ä yLL Š
tan-1 Hx, yL
2
sgnHargHzLL Š sgnItan-1 HReHzL, ImHzLLM
12.02.19.0013.01
12.02.19.0014.01
sgnHargHzLL Š
argHzL
tan-1 HReHzL, ImHzLL
2
Representations through equivalent functions
With related functions
With Re
argHzL Š tan-1 HReHzL, -ä Hz - ReHzLLL
12.02.27.0006.01
With Im
argHzL Š tan-1 Hz - ä ImHzL, ImHzLL
12.02.27.0007.01
argHzL Š tan-1 HReHzL, ImHzLL
12.02.27.0004.01
12.02.27.0005.01
argHzL Š tan-1
ImHzL
ReHzL
; ReHzL > 0
With Abs
12.02.27.0001.01
argHzL Š -ä log
z
z¤
argHzL Š ä HlogH z¤L - logHzLL
12.02.27.0008.01
http://functions.wolfram.com
14
12.02.27.0002.01
cosHargHzLL Š
ReHzL
z¤
12.02.27.0003.01
sinHargHzLL Š
ImHzL
z¤
With Conjugate
12.02.27.0009.01
argHzL Š
1
2
ä HlogHz z L - 2 logHzLL
With Sign
12.02.27.0010.01
argHzL Š -ä logHsgnHzLL
With inverse trigonometric functions
With ArcSin
12.02.27.0011.01
-1
argHzL ‡ sin
ImHzL
z¤
+
Π
1
-
4
z +
z
ä
-z2
z2
1-
z
ä
-
z
-ä z + 2
z
12.02.27.0012.01
-1
argHzL ‡ sin
ReHzL
z¤
+
Π
z2
4
z
1+
ä
-z2
1
-
z
z -
z
-
ä
äz +
z
4 H-1L34
äz
+4
z
With ArcCos
12.02.27.0013.01
argHzL ‡ -cos-1
ImHzL
z¤
+
Π
4
1
-
z
z +
ä
-z2
z
1-
z2
z
ä
-
z
-ä z + 4
12.02.27.0014.01
argHzL ‡ -cos-1
ReHzL
z¤
+
Π
z2
4
z
1+
ä
-z2
z
With ArcTan
argHzL ‡ tan-1 HReHzL, ImHzLL
12.02.27.0015.01
With inverse hyperbolic functions
Inequalities
-
1
z
z -
-
ä
z
äz +
4 H-1L34
z
äz
+6
http://functions.wolfram.com
argHzL¤ £ Π
12.02.29.0001.01
12.02.29.0002.01
-Π < argHzL £ Π
-r < argHa + zL < R ; -r < argHzL < R ì -r < argHaL < R ì R - r £ Π
12.02.29.0003.01
Pavlyk O. (2006)
Zeros
argHzL Š 0 ; z Î R ß z > 0
12.02.30.0001.01
History
Arg is encountered often in mathematics and the natural sciences.
15
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Copyright
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16