Heaviside function

The Heaviside function , also called theta , step , threshold , step , jump or unit jump function, is a function that is often used in mathematics and physics . It is named after the British mathematician and physicist Oliver Heaviside (1850–1925).

General

The Heaviside function has the value zero for any negative number , otherwise the value one . The Heaviside function is continuous everywhere except for the digit . Written in formulas this means: ${\ displaystyle x = 0}$

Heaviside function

{\ displaystyle {\ begin {aligned} \ Theta \ colon \; & \ mathbb {R} \ to \ {0,1 \} \\\ & x \ mapsto {\ begin {cases} 0: & x <0 \\ 1 : & x \ geq 0 \ end {cases}} \ end {aligned}}}

So it is the characteristic function of the interval of nonnegative real numbers . ${\ displaystyle [0, + \ infty)}$

In the specialist literature, notations that differ from these are used instead : ${\ displaystyle \ Theta (x)}$

{\ displaystyle H (x)}

, which is based on the name of Oliver H eaviside.

${\ displaystyle s (x)}$ and according to the designation S prungfunktion. ${\ displaystyle \ sigma (x)}$

{\ displaystyle u (x)}

after the term English u nit step function .

Also is used frequently.

{\ displaystyle \ epsilon (x)}

In systems theory is also used the symbol .

{\ displaystyle 1 (x)}

The function has numerous applications, for example in communications engineering or as a mathematical filter: multiplying each value of any continuous function point by point by the corresponding value of the Heaviside function results in a function that has the value zero to the left ( deterministic function), but to the right of it corresponds to the original function. ${\ displaystyle x = 0}$

Alternative representations

The value of the Heaviside function at this point can also be set as follows. To mark the definition one writes ${\ displaystyle x = 0}$

{\ displaystyle {\ begin {aligned} \ Theta _ {c} \ colon \; & \ mathbb {R} \ to \ mathbb {K} \\\, & x \ mapsto {\ begin {cases} 0: & x <0 \\ c: & x = 0 \\ 1: & x> 0 \ end {cases}} \ end {aligned}}}

with . So it can represent any set as long as it contains 0 and 1. Usually, however, is used. ${\ displaystyle 0,1, c \ in \ mathbb {K}}$ ${\ displaystyle \ mathbb {K}}$ ${\ displaystyle \ mathbb {K} = [0,1] \ subset \ mathbb {R}}$

This definition is characterized by the property that then is. ${\ displaystyle \ Theta _ {c} (0) = c}$

By choosing and consequently one achieves that the equations ${\ displaystyle c: = {\ tfrac {1} {2}}}$ ${\ displaystyle \ Theta _ {\ frac {1} {2}} (0) = \ textstyle {\ frac {1} {2}}}$

{\ displaystyle \ Theta _ {\ frac {1} {2}} (x) = {\ tfrac {1} {2}} (\ operatorname {sgn} {(x)} + 1)}

and so too

{\ displaystyle \ Theta _ {\ frac {1} {2}} (- x) = 1- \ Theta _ {\ frac {1} {2}} (x)}

are valid for all real ones. ${\ displaystyle x}$

An integral representation of the Heaviside step function is as follows:

{\ displaystyle \ Theta (x) = - \ lim _ {\ varepsilon \ to 0} {1 \ over 2 \ pi i} \ int _ {- \ infty} ^ {\ infty} {1 \ over \ tau + i \ varepsilon} e ^ {- ix \ tau} \, \ mathrm {d} \ tau}

Another representation is given by

{\ displaystyle \ Theta (x) = \ lim _ {\ varepsilon \ to 0} {1 \ over \ pi} \ left [\ arctan \ left ({x \ over \ varepsilon} \ right) + {\ pi \ over 2} \ right]}

properties

Differentiability

The Heaviside function is neither differentiable in the classical sense nor is it weakly differentiable . Nevertheless, one can define a derivation via the theory of distributions . The derivative of the Heaviside function in this sense is the Dirac delta distribution , which is used in physics to describe point sources of fields.

{\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ Theta (x) = \ delta (x)}

A heuristic explanation for this formula is obtained if one and suitable approximated z. B. by ${\ displaystyle \ Theta (x)}$ ${\ displaystyle \ delta (x)}$

{\ displaystyle \ Theta _ {\ epsilon} (x): = {\ begin {cases} 0 & x <(- \ epsilon) \\\ left ({\ frac {1} {2}} + {\ frac {x} {2 \ epsilon}} \ right) & | x | \ leq \ epsilon \\ 1 & x> \ epsilon \ ,, \ end {cases}}}

such as

{\ displaystyle \ delta _ {\ epsilon} (x): = {\ begin {cases} 0 & | x |> \ epsilon \\ {\ frac {1} {2 \ epsilon}} & | x | \ leq \ epsilon \ ,, \ end {cases}}}

whereby the limit value is considered in each case . ${\ displaystyle \ lim _ {\ epsilon \ searrow 0}}$

Alternatively, a differentiable approximation to the Heaviside function can be achieved using a correspondingly standardized sigmoid function .

integration

An antiderivative of the Heaviside step function is obtained by splitting the integral according to the two cases and from the case distinction in the definition: ${\ displaystyle x <0}$ ${\ displaystyle x \ geq 0}$

for true ${\ displaystyle x> 0}$
${\ displaystyle {\ begin {aligned} \ int _ {- \ infty} ^ {x} \ Theta \! \ left (t \ right) \, \ mathrm {d} t & = \ int _ {- \ infty} ^ {x} \ left \ {{\ begin {alignedat} {2} 0 & {\ text {,}} & \ quad & {\ text {if}} t <0 \\ 1 & {\ text {,}} && { \ text {falls}} t \ geq 0 \ end {alignedat}} \ right. \, \ mathrm {d} t = \ int _ {- \ infty} ^ {0} \ left \ {{\ begin {alignedat} {2} 0 & {\ text {,}} & \ quad & {\ text {if}} t <0 \\ 1 & {\ text {,}} && {\ text {if}} t \ geq 0 \ end { alignedat}} \ right. \, \ mathrm {d} t + \ int _ {0} ^ {x} \ left \ {{\ begin {alignedat} {2} 0 & {\ text {,}} & \ quad & { \ text {falls}} t <0 \\ 1 & {\ text {,}} && {\ text {falls}} t \ geq 0 \ end {alignedat}} \ right. \, \ mathrm {d} t \\ & = \ int _ {- \ infty} ^ {0} 0 \, \ mathrm {d} t + \ int _ {0} ^ {x} 1 \, \ mathrm {d} t = \ int _ {0} ^ {x} 1 \, \ mathrm {d} t = {\ Big [} t {\ Big]} _ {0} ^ {x} = x \ end {aligned}}}$
In fact, only the first case occurs and it applies ${\ displaystyle x \ leq 0}$
${\ displaystyle \ int _ {- \ infty} ^ {x} \ Theta \! \ left (t \ right) \, \ mathrm {d} t = \ int _ {- \ infty} ^ {x} \ left \ {{\ begin {alignedat} {2} 0 & {\ text {,}} & \ quad & {\ text {if}} t <0 \\ 1 & {\ text {,}} && {\ text {if}} t \ geq 0 \ end {alignedat}} \ right. \, \ mathrm {d} t = \ int _ {- \ infty} ^ {x} 0 \, \ mathrm {d} t = 0}$ .