signum function

signum function

The

signum function

is the

function

sgn

:

ℝ

→

ℝ

sgn

(

x

)

=

{

-

1

when

⁢

x

<

0

,

0

when

⁢

x

=

0

,

1

when

⁢

x

>

0

.

The following properties hold:

1.

For all

x

∈

ℝ

,

sgn

(

-

x

)

=

-

sgn

(

x

)

.

2.

For all

x

∈

ℝ

,

|

x

|

=

sgn

(

x

)

⁢

x

.

3.

For all

x

≠

0

,

d

d

⁢

x

⁢

|

x

|

=

sgn

(

x

)

.

Here, we should point out that the signum function
is often defined simply as

1

for

x

>

0

and

-

1

for

x

<

0

.
Thus, at

x

=

0

, it is left undefined. See for example

[

1

]

.
In applications such as the

Laplace transform

this definition is adequate,
since the value of a function at a single point does not change the

analysis

.
One could then, in fact, set

sgn

(

0

)

to any value.
However, setting

sgn

(

0

)

=

0

is motivated by the above

relations

.
On a related note, we can extend the definition to the

extended real numbers

ℝ

¯

=

ℝ

∪

{

∞

,

-

∞

}

by defining

sgn

(

∞

)

=

1

and

sgn

(

-

∞

)

=

-

1

.

A related function is the

Heaviside step function

defined as

H

⁢

(

x

)

=

{

0

when

⁢

x

<

0

,

1

/

2

when

⁢

x

=

0

,

1

when

⁢

x

>

0

.

Again, this function is sometimes left undefined at

x

=

0

.
The motivation for setting

H

⁢

(

0

)

=

1

/

2

is that
for all

x

∈

ℝ

, we then have the relations

H

⁢

(

x

)

=

1

2

⁢

(

sgn

(

x

)

+

1

)

,

H

⁢

(

-

x

)

=

1

-

H

⁢

(

x

)

.

This first relation is clear. For the second, we have

1

-

H

⁢

(

x

)

=

1

-

1

2

⁢

(

sgn

(

x

)

+

1

)

=

1

2

⁢

(

1

-

sgn

(

x

)

)

=

1

2

⁢

(

1

+

sgn

(

-

x

)

)

=

H

⁢

(

-

x

)

.

Example

Let

a

<

b

be

real numbers

, and let

f

:

ℝ

→

ℝ

be the

piecewise

defined function

f

⁢

(

x

)

=

{

4

when

⁢

x

∈

(

a

,

b

)

,

0

otherwise.

Using the Heaviside step function, we can write

f

⁢

(

x

)

=

4

⁢

(

H

⁢

(

x

-

a

)

-

H

⁢

(

x

-

b

)

)

(1)

almost everywhere.
Indeed, if we calculate

f

using equation

1

we obtain

f

⁢

(

x

)

=

4

for

x

∈

(

a

,

b

)

,

f

⁢

(

x

)

=

0

for

x

∉

[

a

,

b

]

,
and

f

⁢

(

a

)

=

f

⁢

(

b

)

=

2

. Therefore, equation

1

holds at all points except

a

and

b

.

□

1

Signum function for complex arguments

For a

complex number

z

, the signum function is defined as

[

2

]

sgn

(

z

)

=

{

0

when

⁢

z

=

0

,

z

/

|

z

|

when

⁢

z

≠

0

.

In other words, if

z

is non-zero, then

sgn

z

is the projection
of

z

onto the

unit circle

{

z

∈

ℂ

∣

|

z

|

=

1

}

.
Clearly, the

complex

signum function reduces to the real signum function
for real

arguments

.
For all

z

∈

ℂ

, we have

z

⁢

sgn

z

¯

=

|

z

|

,

where

z

¯

is the

complex conjugate

of

z

.

References

1

E. Kreyszig,

Advanced Engineering Mathematics

,
John Wiley & Sons, 1993, 7th ed.

2

G. Bachman, L. Narici,

Functional analysis

, Academic Press, 1966.

Title

signum function

Canonical name

SignumFunction

Date of creation

2013-03-22 13:36:41

Last modified on

2013-03-22 13:36:41

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 30-00

Classification

msc 26A06

Related topic

ModulusOfComplexNumber

Related topic

HeavisideStepFunction

Related topic

PlusSign

Related topic

SineIntegralInInfinity

Related topic

ListOfImproperIntegrals

Defines

Heavyside step function

Defines

step function

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