The sign function or signum function (from Latin signum 'sign' ) is a function in mathematics that assigns its sign to a real or complex number .
Sign function on the real numbers
definition
The real sign function maps the set of real numbers into the set and is usually defined as follows:
{
-
1
,
0
,
1
}
{\ displaystyle \ {- 1,0,1 \}}
so-called
(
x
)
: =
{
+
1
if
x
>
0
0
if
x
=
0
-
1
if
x
<
0
{\ displaystyle \ operatorname {sgn} (x): = {\ begin {cases} +1 & \; {\ text {falls}} \ quad x> 0 \\\; \; \, 0 & \; {\ text { if}} \ quad x = 0 \\ - 1 & \; {\ text {falls}} \ quad x <0 \\\ end {cases}}}
She assigns the positive numbers the value +1, the negative numbers the value −1 and the 0 the value 0.
In computing technology applications, the special position of 0 is sometimes dispensed with; this is then assigned to the positive, negative or both number ranges. This allows the sign of a number to be encoded in a single bit . Since the zero is a zero set below the Lebesgue measure , this is not important for all practical applications.
In the event that is set, there is the following relationship to the Heaviside function :
so-called
(
0
)
=
1
{\ displaystyle \ operatorname {sgn} (0) = 1}
Θ
(
x
)
{\ displaystyle \ Theta (x)}
so-called
(
x
)
=
2
Θ
(
x
)
-
1
{\ displaystyle \ operatorname {sgn} (x) = 2 \ Theta (x) -1}
Calculation rules
By differentiating between cases it is easy to prove:
For everyone with amount applies .
x
∈
R.
{\ displaystyle x \ in \ mathbb {R}}
|
x
|
{\ displaystyle | x |}
x
=
|
x
|
⋅
so-called
(
x
)
such as
x
⋅
so-called
(
x
)
=
|
x
|
{\ displaystyle x = | x | \ cdot \ operatorname {sgn} (x) \; {\ text {and}} \; x \ cdot \ operatorname {sgn} (x) = | x |}
so-called
(
-
x
)
=
-
so-called
(
x
)
{\ displaystyle \ operatorname {sgn} (- x) = - \ operatorname {sgn} (x)}
for everyone .
x
∈
R.
{\ displaystyle x \ in \ mathbb {R}}
If is a constant and an odd function , then
k
∈
R.
{\ displaystyle k \ in \ mathbb {R}}
f
{\ displaystyle f}
f
(
k
⋅
x
)
=
f
(
so-called
(
k
)
⋅
|
k
|
⋅
x
)
=
so-called
(
k
)
⋅
f
(
|
k
|
⋅
x
)
{\ displaystyle f (k \ cdot x) \ quad = f (\ operatorname {sgn} (k) \ cdot | k | \ cdot x) \ quad = \ operatorname {sgn} (k) \ cdot f (| k | \ cdot x)}
For the transition to the reciprocal number is compatible with the signum function and does not change its value:
x
≠
0
{\ displaystyle x \ neq 0}
so-called
(
x
-
1
)
=
(
so-called
(
x
)
)
-
1
=
so-called
(
x
)
{\ displaystyle \ operatorname {sgn} (x ^ {- 1}) = (\ operatorname {sgn} (x)) ^ {- 1} = \ operatorname {sgn} (x)}
for everyone .
0
≠
x
∈
R.
{\ displaystyle 0 \ neq x \ in \ mathbb {R}}
The signum function is compatible with multiplication:
so-called
(
x
)
⋅
so-called
(
y
)
=
so-called
(
x
⋅
y
)
{\ displaystyle \ operatorname {sgn} (x) \ cdot \ operatorname {sgn} (y) = \ operatorname {sgn} (x \ cdot y)}
for everyone .
x
,
y
∈
R.
{\ displaystyle x, y \ in \ mathbb {R}}
so-called
(
so-called
(
x
)
)
=
so-called
(
x
)
{\ displaystyle \ operatorname {sgn} (\ operatorname {sgn} (x)) = \ operatorname {sgn} (x)}
for everyone .
x
∈
R.
{\ displaystyle x \ in \ mathbb {R}}
From the last two calculation rules mentioned, it follows, for example, that the signum function in an argument composed of any number of factors can be replaced by a factor without changing the function value:
x
j
{\ displaystyle x_ {j}}
so-called
(
x
j
)
{\ displaystyle \ operatorname {sgn} (x_ {j})}
so-called
(
x
j
⋅
∏
i
x
i
)
=
so-called
(
so-called
(
x
j
)
⋅
∏
i
x
i
)
{\ displaystyle \ operatorname {sgn} {\ bigg (} x_ {j} \ cdot \ prod _ {i} x_ {i} {\ bigg)} = \ operatorname {sgn} {\ bigg (} \ operatorname {sgn} (x_ {j}) \ cdot \ prod _ {i} x_ {i} {\ bigg)}}
for any .
x
i
,
x
j
∈
R.
{\ displaystyle x_ {i}, x_ {j} \ in \ mathbb {R}}
Derivative and integral
The sign function is not continuous at the 0 position.
The sign function is not continuous at this point and therefore not classically differentiable there. For all other places the sign function can be differentiated with . The sign function also has no weak derivative . However, it is differentiable in the sense of distributions , and its derivative is , where denotes the delta distribution .
x
=
0
{\ displaystyle x = 0}
x
≠
0
{\ displaystyle x \ neq 0}
so-called
′
(
x
)
=
0
{\ displaystyle \ operatorname {sgn} ^ {\ prime} (x) = 0}
2
δ
{\ displaystyle 2 \ delta}
δ
{\ displaystyle \ delta}
Furthermore applies to everyone
x
∈
R.
{\ displaystyle x \ in \ mathbb {R}}
|
x
|
=
∫
0
x
so-called
(
t
)
d
t
.
{\ displaystyle | x | = \ int _ {0} ^ {x} \ operatorname {sgn} (t) \, dt \ ,.}
The sign function is also the weak derivative of the absolute value function .
Sign function on the complex numbers
definition
Signum of four complex numbers
Compared to the sign function of real numbers, the following extension to complex numbers is rarely considered:
so-called
(
z
)
: =
{
z
|
z
|
if
z
≠
0
0
if
z
=
0
{\ displaystyle \ operatorname {sgn} (z): = {\ begin {cases} {\ frac {z} {| z |}} & \; {\ text {if}} \ quad z \ neq 0 \\ 0 & \; {\ text {falls}} \ quad z = 0 \\\ end {cases}}}
The result of this function is on the unit circle and has the same argument as the output value, in particular applies
z
≠
0
{\ displaystyle z \ neq 0}
so-called
(
r
e
i
φ
)
=
e
i
φ
,
f
a
l
l
s
r
>
0.
{\ displaystyle \ operatorname {sgn} \ left (r \ mathrm {e} ^ {\ mathrm {i} \ varphi} \ right) = \ mathrm {e} ^ {\ mathrm {i} \ varphi}, \ qquad \ mathrm {if} \ r> 0.}
Example: (red in the picture)
z
1
=
2
+
2
i
{\ displaystyle z_ {1} = 2 + 2 \ mathrm {i}}
so-called
(
z
1
)
=
so-called
(
2
+
2
i
)
=
2
+
2
i
|
2
+
2
i
|
=
2
+
2
i
2
2
=
1
+
i
2
=
2
2
+
2
2
i
.
{\ displaystyle \ operatorname {sgn} (z_ {1}) = \ operatorname {sgn} (2 + 2 \ mathrm {i}) = {\ frac {2 + 2 \ mathrm {i}} {\ left | 2+ 2 \ mathrm {i} \ right |}} = {\ frac {2 + 2 \ mathrm {i}} {2 {\ sqrt {2}}}} = {\ frac {1+ \ mathrm {i}} { \ sqrt {2}}} = {\ frac {\ sqrt {2}} {2}} + {\ frac {\ sqrt {2}} {2}} {\ mathrm {i}}.}
Calculation rules
The following calculation rules apply to the complex sign function:
For all complex numbers and the following applies:
z
{\ displaystyle z}
w
{\ displaystyle w}
z
=
|
z
|
⋅
so-called
z
{\ displaystyle z = | z | \ cdot \ operatorname {sgn} z}
for all where denotes the amount of ;
z
,
{\ displaystyle z,}
|
z
|
{\ displaystyle | z |}
z
{\ displaystyle z}
so-called
(
z
¯
)
=
so-called
(
z
)
¯
{\ displaystyle \ operatorname {sgn} ({\ bar {z}}) = {\ overline {\ operatorname {sgn} (z)}}}
where the slash denotes the complex conjugation ;
so-called
(
z
⋅
w
)
=
so-called
z
⋅
so-called
w
{\ displaystyle \ operatorname {sgn} (z \ cdot w) = \ operatorname {sgn} z \ cdot \ operatorname {sgn} w}
, in particular
so-called
(
λ
⋅
z
)
=
so-called
z
{\ displaystyle \ operatorname {sgn} (\ lambda \ cdot z) = \ operatorname {sgn} z}
for positive real ones ,
λ
{\ displaystyle \ lambda}
so-called
(
λ
⋅
z
)
=
-
so-called
z
{\ displaystyle \ operatorname {sgn} (\ lambda \ cdot z) = - \ operatorname {sgn} z}
for negative reals ,
λ
{\ displaystyle \ lambda}
so-called
(
-
z
)
=
-
so-called
(
z
)
{\ displaystyle \ operatorname {sgn} (-z) = - \ operatorname {sgn} (z)}
;
so-called
(
|
z
|
)
=
|
so-called
(
z
)
|
{\ displaystyle \ operatorname {sgn} (| z |) = | \ operatorname {sgn} (z) |}
.
If is, also applies
z
≠
0
{\ displaystyle z \ neq 0}
so-called
(
z
-
1
)
=
so-called
(
z
)
-
1
=
so-called
(
z
)
¯
{\ displaystyle \ operatorname {sgn} (z ^ {- 1}) = \ operatorname {sgn} (z) ^ {- 1} = {\ overline {\ operatorname {sgn} (z)}}}
.
literature
Königsberger: Analysis 1 . 6th edition. Springer, Berlin 2003, ISBN 3-540-40371-X , p. 101 .
Hildebrandt: Analysis 1 . 2nd Edition. Springer, Berlin 2005, ISBN 3-540-25368-8 , pp. 133 .
Web links
<img src="https://de.wikipedia.org//de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">