HeavisideTheta

Notations

Traditional name

Heaviside step function

Traditional notation

Θ x

Mathematica StandardForm notation H L HeavisideTheta x

Primary definition@ D

14.05.02.0001.01 Θ x ‡ 1 ; x Î R x > 0

14.05.02.0002.01 ΘHxL ‡ 0 ; x Î R ß x < 0

14.05.02.0003.01 ¶Θ x H L ‡∆ x ß ¶x

H L Specific HvaluesL

Values at fixed points

14.05.03.0001.01 Θ 1 ‡ 1

14.05.03.0002.01 ΘH-L1 ‡ 0

Values at infinities H L 14.05.03.0003.01 Θ ¥ ‡ 1

14.05.03.0004.01 ΘH-¥L ‡ 0

H L http://functions.wolfram.com 2

General characteristics

Domain and analyticity

Θ x is a nonanalytical function; it is a piecewise constant function defined for all nonzero real x.

14.05.04.0001.01 x™Θ x › R™Z H L Symmetries and periodicities H L Parity

14.05.04.0002.01 Θ -x Š 1 - Θ x ; x ¹ 0

Periodicity H L H L  No periodicity

Sets of discontinuity

The function Θ x is continuous function in R \ 0 .

14.05.04.0003.01 DS Θ x Š 0 x H L 8 < 14.05.04.0004.01 lim Θ Ε Š 1 Ε®+0 H H LL 8 <

14.05.04.0005.01 lim ΘH-LΕ Š 0 Ε®+0

SeriesH representationsL

Exponential Fourier series

14.05.06.0001.01 2 ¥ sin 2 k + 1 x 1 Θ x Š + ; -Π < x < Π Π k=0 2 k + 1 2 HH L L ResidueH L â representations

14.05.06.0002.01 1 -s Θ x Š ress 1 - x 0 ; 0 < x < 2 s

14.05.06.0003.01 ΘHxL Š 0 ; xH< 0 L H L 

H L  http://functions.wolfram.com 3

14.05.06.0004.01 1 -s Θ x Š ress 1 + x 0 ; x > 0 s

14.05.06.0005.01 ΘHxL Š 0 ; -H2 < xL< 0 H L 

IntegralH L  representations

On the real axis

Of the direct function

14.05.07.0001.01 1 ¥ ãä t x Θ x Š lim ât 2 Π ä ¶®+0 -¥ t - ä ¶

14.05.07.0002.01 H L 1 à ¥ ã-ä t x Θ x Š lim ât 2 Π ä ¶®+0 -¥ t + ä ¶

14.05.07.0003.01

H L t2 1 à ¥ 1 - Θ x Š lim ã ¶2 ât Π ¶®+0 -x ¶

14.05.07.0004.01 H L 1 xà1 t Θ x Š lim sin ât Π ¶®+0 -¥ t ¶

ContourH L integralà representations

14.05.07.0005.01 1 Γ+ä ¥ G s 1 - x -s Θ x Š âs ; 0 < Γ x < 2 2 Π ä Γ-ä ¥ G s + 1

14.05.07.0006.01H L H L H L 1 à G s 1 - x -s  ì Θ x Š H âL s ; x < 2 2 Π ä L G s + 1

14.05.07.0007.01H L H L H L 1 à Γ+ä ¥ G -s 1 + x - s Θ x Š H L âs ; Γ < 0 x > -2 2 Π ä Γ-ä ¥ G 1 - s

14.05.07.0008.01H L H L H L 1 à G -s 1 + x -s  ì Θ x Š H L âs ; x > -2 2 Π ä L G 1 - s

H L H L OtherH L integralà representations H L 14.05.07.0009.01 Θ x ‡ ∆ x â x

H L à H L http://functions.wolfram.com 4

Limit representations

14.05.09.0001.01 1 Θ x Š lim x ¶®+0 - ã ¶ + 1

H L 14.05.09.0002.01 x - Θ x Š lim ãã ¶ ¶®+0

14.05.09.0003.01 H L 1 x Θ x Š lim tanh + 1 2 ¶®+0 ¶

14.05.09.0004.01 H L 1 Θ x Š lim log ä ¶ - x - log -x - ä ¶ 2 Π ä ¶®+0

14.05.09.0005.01 H L 1 H xH 1 L H LL Θ x Š lim tan-1 + Π ¶®+0 ¶ 2

14.05.09.0006.01 H L 1 x Θ x Š lim erf + 1 2 ¶®+0 ¶

14.05.09.0007.01 H L 1 x Θ x Š lim erfc - 2 ¶®+0 ¶

14.05.09.0008.01 H L 1 Π x 1 Θ x Š lim Si + Π ¶®+0 ¶ 2

14.05.09.0009.01 H L ¶ + ä x Λ ãä Λ Π Θ x -Θ -x Š lim ¶®+0 ¶ - ä x H H L H LL Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

14.05.16.0001.01 Θ -x Š 1 - Θ x ; x ¹ 0

Multiple arguments H L H L  14.05.16.0002.01 n n n - 1 n-k Θ x - ak Š -1 Θ x - ak + n - 2 - 1 ; x ¹ ak ak < ak+1 ak Î R 1 £ k £ n k=1 k=1 2

äH L âH L H L  ì ì ì http://functions.wolfram.com 5

Products, sums, and powers of the direct function

Powers of the direct function

14.05.16.0003.01 Θ x a Š Θ x ; a > 0

IdeHnLtitieHsL 

Functional identities

14.05.17.0001.01 b b Θ a x + b ‡ Θ - - x Θ -a + Θ + x Θ a ; a Î R b Î R a a

14.05.17.0002.01 H L x + yH L H L  ì Θ x Θ y ‡ Θ x y Θ ; y Î R y Î R 2

ComplexH L H L HcharacteristicsL  ì

Real part

14.05.19.0001.01 Re Θ x ‡ Θ x

Imaginary part H H LL H L 14.05.19.0002.01 Im Θ x ‡ 0

Absolute value H H LL 14.05.19.0003.01 Θ x ‡ Θ x

Argument H L¤ H L 14.05.19.0004.01 arg Θ x ‡ tan-1 Θ x , 0

Conjugate value H H LL H H L L 14.05.19.0005.01 Θ x Š Θ x

DifHfeL renH tL iation

Low-order differentiation

In a distributional sense for x Î R . http://functions.wolfram.com 6

14.05.20.0001.01 ¶Θ x ‡ ∆ x ¶x

H L IntegratiHoLn

Indefinite integration

Involving only one direct function

14.05.21.0001.01 Θ x â x ‡ x Θ x

Involving one direct function and elementary functions à H L H L Involving power function

14.05.21.0002.01 xΑ xΑ-1 Θ x â x ‡ Θ x Α

14.05.21.0003.01 à Θ x H L H L â x ‡ log x Θ x x

H L Definiteà integrationH L H L

14.05.21.0004.01 ¥ Θ t Θ x - t ât ‡ x Θ x -¥

Integralà H L H transformsL H L

Fourier exp transforms

14.05.22.0001.01 ä Π Ft Θ t x ‡ + ∆ x 2 Π x 2

Inverse@ H LD H LFourier exp transformsH L

14.05.22.0002.01 ä Π -1 Ft Θ t x ‡ - + ∆ x 2 Π x 2

Fourier@ H LD cosH L transforms H L

14.05.22.0003.01 Π Fct Θ t x Š ∆ x 2

@ H LD H L H L http://functions.wolfram.com 7

Fourier sin transforms

14.05.22.0004.01 2 1 Fst Θ t z Š Π z

Laplace@ H LD H transformsL

14.05.22.0005.01 1 Lt Θ t z ‡ z

Representations@ H LD H L through more general functions

Through Meijer G

Classical cases for the direct function itself

14.05.26.0001.01 1 Θ x ‡ G1,0 1 - x ; x < 2 1,1 0

14.05.26.0002.01 H L 1  Θ x ‡ G0,1 x + 1 ; x > -2 1,1 0

14.05.26.0003.01 H L 1  Θ 1 - z ‡ G1,0 z 1,1 0

14.05.26.0004.01 H ¤L 1 Θ z - 1 ‡ G0,1 z 1,1 0

RepresentationsH ¤ L through equivalent functions

14.05.27.0001.01

Θ x ‡ Θ x1, x2, ¼, xn ; x1 ‡ x n ‡ 1

14.05.27.0002.01 1 ΘHxL ‡ H sgn x + 1 L;x Î R xì¹ 0 2

14.05.27.0003.01

ΘHxL ‡ Θ xH1, x2H, L¼, xLn ; x1 Šßx ¹ 0 n ‡ 1

14.05.27.0004.01 ΘHxL ‡ ΘHx ; x ¹ 0 L  ì Heaviside theta function Θ x represents a generalized function,and unit step function Θ x represents a piecewise function.H L H LThey coincide almost everywhere: Θ x ‡ Θ x ; x ¹ 0.

H L H L History H L H L  http://functions.wolfram.com 8

Ð O. Heaviside (1881) http://functions.wolfram.com 9

Copyright

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