Functions
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Symbolic Functions
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Dirac Delta Functions
Dirac Delta Functions
•
Δ(x)
—Returns the Dirac delta (Unit Impulse) function, which is defined as follows:
<region id="ID0E6PWBB" top="19.200000000000003" left="19.200000000000003" actualWidth="66.353333333333339" actualHeight="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="5"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">Δ</id> <id xml:space="preserve">x</id> </apply> <real>0</real> </apply> </math> </region>
if
<region id="ID0EBRWBB" top="19.200000000000003" left="230.40000000000003" actualWidth="37.176666666666662" actualHeight="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="6"> <apply xmlns="http://schemas.mathsoft.com/math50"> <notEqual /> <id xml:space="preserve">x</id> <real>0</real> </apply> </math> </region>
and
<region id="ID0EYSWBB" top="76.800000000000011" left="19.200000000000003" actualWidth="104.39833333333335" actualHeight="56.800000000000004" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="7"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <integral /> <lambda> <boundVars> <id xml:space="preserve">x</id> </boundVars> <apply> <id xml:space="preserve">Δ</id> <id xml:space="preserve">x</id> </apply> </lambda> <lowerBound> <apply> <neg /> <id xml:space="preserve">∞</id> </apply> </lowerBound> <upperBound> <id xml:space="preserve">∞</id> </upperBound> </apply> <real>1</real> </apply> </math> </region>
•
Δ(n, x)
—Returns the
n
th
derivative of the Dirac delta function, which is defined as follows:
<region id="ID0EEVWBB" top="153.60000000000002" left="19.200000000000003" actualWidth="184.83833333333337" actualHeight="53.743333445231123" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="8"> <symEval xmlns="http://schemas.mathsoft.com/math50"> <apply> <derivative /> <lambda> <boundVars> <id xml:space="preserve">τ</id> </boundVars> <apply> <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">Δ</id> <sequence> <id xml:space="preserve">n</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">τ</id> </sequence> </apply> </lambda> <degree> <placeholder /> </degree> </apply> <command> <placeholder /> </command> <symResult> <apply> <id labels="FUNCTION" xml:space="preserve">Δ</id> <sequence> <apply> <plus /> <id labels="*" xml:space="preserve">n</id> <real>1</real> </apply> <id labels="VARIABLE" xml:space="preserve">τ</id> </sequence> </apply> </symResult> </symEval> </math> </region>
Δ(x)
relates to
Δ(n, x)
as follows:
<region id="ID0EAWWBB" top="230.40000000000003" left="19.200000000000003" actualWidth="109.92666666666668" actualHeight="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="9"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">Δ</id> <id xml:space="preserve">x</id> </apply> <apply> <id xml:space="preserve">Δ</id> <sequence> <real>0</real> <id xml:space="preserve">x</id> </sequence> </apply> </apply> </math> </region>
Arguments
•
x
is a real scalar.
•
n
is a nonnegative integer.
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