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).
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
${\ displaystyle {\ begin {aligned} \ Theta \ colon \; & \ mathbb {R} \ to \ {0,1 \} \\\ & x \ mapsto {\ begin {cases} 0: & x <0 \\ 1 : & x \ geq 0 \ end {cases}} \ end {aligned}}}$
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}}}$${\ displaystyle \ Theta _ {\ frac {1} {2}} (0) = \ textstyle {\ frac {1} {2}}}$
${\ displaystyle \ Theta _ {\ frac {1} {2}} (x) = {\ tfrac {1} {2}} (\ operatorname {sgn} {(x)} + 1)}$ and so too
${\ displaystyle \ Theta (x) = - \ lim _ {\ varepsilon \ to 0} {1 \ over 2 \ pi i} \ int _ {- \ infty} ^ {\ infty} {1 \ over \ tau + i \ varepsilon} e ^ {- ix \ tau} \, \ mathrm {d} \ tau}$
Another representation is given by
${\ displaystyle \ Theta (x) = \ lim _ {\ varepsilon \ to 0} {1 \ over \ pi} \ left [\ arctan \ left ({x \ over \ varepsilon} \ right) + {\ pi \ over 2} \ right]}$
properties
Differentiability
The Heaviside function is neither differentiable in the classical sense nor is it weakly differentiable . Nevertheless, one can define a derivation via the theory of distributions . The derivative of the Heaviside function in this sense is the Dirac delta distribution , which is used in physics to describe point sources of fields.
A heuristic explanation for this formula is obtained if one and suitable approximated z. B. by
${\ displaystyle \ Theta (x)}$${\ displaystyle \ delta (x)}$
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
${\ displaystyle x \ leq 0}$
${\ displaystyle \ int _ {- \ infty} ^ {x} \ Theta \! \ left (t \ right) \, \ mathrm {d} t = \ int _ {- \ infty} ^ {x} \ left \ {{\ begin {alignedat} {2} 0 & {\ text {,}} & \ quad & {\ text {if}} t <0 \\ 1 & {\ text {,}} && {\ text {if}} t \ geq 0 \ end {alignedat}} \ right. \, \ mathrm {d} t = \ int _ {- \ infty} ^ {x} 0 \, \ mathrm {d} t = 0}$.
Taken together, then
${\ displaystyle \ int _ {- \ infty} ^ {x} \ Theta \! \ left (t \ right) \, \ mathrm {d} t = \ left \ {{\ begin {alignedat} {2} 0 & { \ text {,}} & \ quad & {\ text {if}} x \ leq 0 \\ x & {\ text {,}} && {\ text {if}} x> 0 \ end {alignedat}} \ right .}$
respectively
${\ displaystyle \ int _ {- \ infty} ^ {x} \ Theta \! \ left (t \ right) \, \ mathrm {d} t = \ max \ left \ {0, x \ right \}}$.
The set of all antiderivatives of the Heaviside function is thus
${\ displaystyle \ int \ Theta \! \ left (t \ right) \, \ mathrm {d} t = \ max \ left \ {0, x \ right \} + C}$.