# Sign function

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: ${\ displaystyle \ {- 1,0,1 \}}$ ${\ displaystyle \ operatorname {sgn} (x): = {\ begin {cases} +1 & \; {\ text {falls}} \ quad x> 0 \\\; \; \, 0 & \; {\ text { if}} \ quad x = 0 \\ - 1 & \; {\ text {falls}} \ quad x <0 \\\ end {cases}}}$ 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 : ${\ displaystyle \ operatorname {sgn} (0) = 1}$ ${\ displaystyle \ Theta (x)}$ ${\ displaystyle \ operatorname {sgn} (x) = 2 \ Theta (x) -1}$ ### Calculation rules

By differentiating between cases it is easy to prove:

• For everyone with amount applies .${\ displaystyle x \ in \ mathbb {R}}$ ${\ displaystyle | x |}$ ${\ displaystyle x = | x | \ cdot \ operatorname {sgn} (x) \; {\ text {and}} \; x \ cdot \ operatorname {sgn} (x) = | x |}$ ${\ displaystyle \ operatorname {sgn} (- x) = - \ operatorname {sgn} (x)}$ for everyone .${\ displaystyle x \ in \ mathbb {R}}$ • If is a constant and an odd function , then${\ displaystyle k \ in \ mathbb {R}}$ ${\ displaystyle f}$ ${\ displaystyle f (k \ cdot x) \ quad = f (\ operatorname {sgn} (k) \ cdot | k | \ cdot x) \ quad = \ operatorname {sgn} (k) \ cdot f (| k | \ cdot x)}$ • For the transition to the reciprocal number is compatible with the signum function and does not change its value:${\ displaystyle x \ neq 0}$ ${\ displaystyle \ operatorname {sgn} (x ^ {- 1}) = (\ operatorname {sgn} (x)) ^ {- 1} = \ operatorname {sgn} (x)}$ for everyone .${\ displaystyle 0 \ neq x \ in \ mathbb {R}}$ • The signum function is compatible with multiplication:
${\ displaystyle \ operatorname {sgn} (x) \ cdot \ operatorname {sgn} (y) = \ operatorname {sgn} (x \ cdot y)}$ for everyone .${\ displaystyle x, y \ in \ mathbb {R}}$ ${\ displaystyle \ operatorname {sgn} (\ operatorname {sgn} (x)) = \ operatorname {sgn} (x)}$ for everyone .${\ displaystyle x \ in \ mathbb {R}}$ 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: ${\ displaystyle x_ {j}}$ ${\ displaystyle \ operatorname {sgn} (x_ {j})}$ ${\ displaystyle \ operatorname {sgn} {\ bigg (} x_ {j} \ cdot \ prod _ {i} x_ {i} {\ bigg)} = \ operatorname {sgn} {\ bigg (} \ operatorname {sgn} (x_ {j}) \ cdot \ prod _ {i} x_ {i} {\ bigg)}}$ for any .${\ displaystyle x_ {i}, x_ {j} \ in \ mathbb {R}}$ ### Derivative and integral

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 . ${\ displaystyle x = 0}$ ${\ displaystyle x \ neq 0}$ ${\ displaystyle \ operatorname {sgn} ^ {\ prime} (x) = 0}$ ${\ displaystyle 2 \ delta}$ ${\ displaystyle \ delta}$ Furthermore applies to everyone ${\ displaystyle x \ in \ mathbb {R}}$ ${\ displaystyle | x | = \ int _ {0} ^ {x} \ operatorname {sgn} (t) \, dt \ ,.}$ The sign function is also the weak derivative of the absolute value function .

## Sign function on the complex numbers

### definition

Compared to the sign function of real numbers, the following extension to complex numbers is rarely considered:

${\ displaystyle \ operatorname {sgn} (z): = {\ begin {cases} {\ frac {z} {| z |}} & \; {\ text {if}} \ quad z \ neq 0 \\ 0 & \; {\ text {falls}} \ quad z = 0 \\\ end {cases}}}$ The result of this function is on the unit circle and has the same argument as the output value, in particular applies ${\ displaystyle z \ neq 0}$ ${\ displaystyle \ operatorname {sgn} \ left (r \ mathrm {e} ^ {\ mathrm {i} \ varphi} \ right) = \ mathrm {e} ^ {\ mathrm {i} \ varphi}, \ qquad \ mathrm {if} \ r> 0.}$ Example: (red in the picture) ${\ displaystyle z_ {1} = 2 + 2 \ mathrm {i}}$ ${\ displaystyle \ operatorname {sgn} (z_ {1}) = \ operatorname {sgn} (2 + 2 \ mathrm {i}) = {\ frac {2 + 2 \ mathrm {i}} {\ left | 2+ 2 \ mathrm {i} \ right |}} = {\ frac {2 + 2 \ mathrm {i}} {2 {\ sqrt {2}}}} = {\ frac {1+ \ mathrm {i}} { \ sqrt {2}}} = {\ frac {\ sqrt {2}} {2}} + {\ frac {\ sqrt {2}} {2}} {\ mathrm {i}}.}$ ### Calculation rules

The following calculation rules apply to the complex sign function:

For all complex numbers and the following applies: ${\ displaystyle z}$ ${\ displaystyle w}$ • ${\ displaystyle z = | z | \ cdot \ operatorname {sgn} z}$ for all where denotes the amount of ;${\ displaystyle z,}$ ${\ displaystyle | z |}$ ${\ displaystyle z}$ • ${\ displaystyle \ operatorname {sgn} ({\ bar {z}}) = {\ overline {\ operatorname {sgn} (z)}}}$ where the slash denotes the complex conjugation ;
• ${\ displaystyle \ operatorname {sgn} (z \ cdot w) = \ operatorname {sgn} z \ cdot \ operatorname {sgn} w}$ , in particular
• ${\ displaystyle \ operatorname {sgn} (\ lambda \ cdot z) = \ operatorname {sgn} z}$ for positive real ones ,${\ displaystyle \ lambda}$ • ${\ displaystyle \ operatorname {sgn} (\ lambda \ cdot z) = - \ operatorname {sgn} z}$ for negative reals ,${\ displaystyle \ lambda}$ • ${\ displaystyle \ operatorname {sgn} (-z) = - \ operatorname {sgn} (z)}$ ;
• ${\ displaystyle \ operatorname {sgn} (| z |) = | \ operatorname {sgn} (z) |}$ .
• If is, also applies${\ displaystyle z \ neq 0}$ ${\ displaystyle \ operatorname {sgn} (z ^ {- 1}) = \ operatorname {sgn} (z) ^ {- 1} = {\ overline {\ operatorname {sgn} (z)}}}$ .

## 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 .