Example: Dirac Delta (Unit Impulse) and LambertW Functions

Functions > Symbolic Functions > Example: Dirac Delta (Unit Impulse) and LambertW Functions

This example provides information about the Dirac delta function and the LambertW function properties.

The Dirac Delta Function

The Dirac delta function delta has the following properties:

The following transforms return results in terms of Dirac delta:

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    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">Φ</id>
        <id xml:space="preserve">τ</id>
      </apply>
      <command>
        <sequence>
          <id xml:space="preserve">fourier</id>
          <id xml:space="preserve">τ</id>
        </sequence>
      </command>
      <symResult>
        <apply>
          <div />
          <apply>
            <plus />
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                <mult />
                <id xml:space="preserve" labels="*">ω</id>
                <id xml:space="preserve" labels="CONSTANT">π</id>
              </apply>
              <apply>
                <id xml:space="preserve" labels="FUNCTION">Δ</id>
                <id xml:space="preserve" labels="*">ω</id>
              </apply>
            </apply>
            <imag symbol="i">1</imag>
          </apply>
          <id xml:space="preserve" labels="*">ω</id>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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In the above example, Φ is the Heaviside step function.

The LambertW Function

The LambertW function appears in the symbolic solution of the following equations:

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      <apply>
        <equal />
        <apply>
          <minus />
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            <real>5</real>
            <id xml:space="preserve">u</id>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">e</id>
            <id xml:space="preserve">u</id>
          </apply>
        </apply>
        <real>0</real>
      </apply>
      <command>
        <sequence>
          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">u</id>
        </sequence>
      </command>
      <symResult>
        <apply>
          <neg />
          <apply>
            <id xml:space="preserve" labels="*">LambertW</id>
            <apply>
              <neg />
              <apply>
                <div />
                <real>1</real>
                <real>5</real>
              </apply>
            </apply>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
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            <id xml:space="preserve">b</id>
            <id xml:space="preserve">v</id>
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        <apply>
          <mult />
          <id xml:space="preserve">c</id>
          <apply>
            <pow />
            <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">e</id>
            <apply>
              <neg />
              <id xml:space="preserve">v</id>
            </apply>
          </apply>
        </apply>
      </apply>
      <command>
        <sequence>
          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">v</id>
        </sequence>
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      <symResult>
        <apply>
          <div />
          <apply>
            <minus />
            <apply>
              <mult />
              <id xml:space="preserve" labels="*">b</id>
              <apply>
                <id xml:space="preserve" labels="*">LambertW</id>
                <apply>
                  <div />
                  <apply>
                    <mult />
                    <id xml:space="preserve" labels="*">c</id>
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="CONSTANT">e</id>
                      <apply>
                        <div />
                        <id xml:space="preserve" labels="*">a</id>
                        <id xml:space="preserve" labels="*">b</id>
                      </apply>
                    </apply>
                  </apply>
                  <id xml:space="preserve" labels="*">b</id>
                </apply>
              </apply>
            </apply>
            <id xml:space="preserve" labels="*">a</id>
          </apply>
          <id xml:space="preserve" labels="*">b</id>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

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        <equal />
        <apply>
          <minus />
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            <apply>
              <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">exp</id>
              <id xml:space="preserve">x</id>
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          <apply>
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            <real>15</real>
            <real>2</real>
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        <real>0</real>
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          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">x</id>
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      <symResult>
        <apply>
          <id xml:space="preserve" labels="*">LambertW</id>
          <apply>
            <div />
            <real>15</real>
            <real>2</real>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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These are the integral, the derivative, and the Taylor series expansion of the LambertW function:

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            <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">LambertW</id>
            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
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          <plus />
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            <mult />
            <id xml:space="preserve" labels="VARIABLE">x</id>
            <apply>
              <id xml:space="preserve" labels="*">LambertW</id>
              <id xml:space="preserve" labels="VARIABLE">x</id>
            </apply>
          </apply>
          <parens>
            <apply>
              <minus />
              <apply>
                <div />
                <id xml:space="preserve" labels="VARIABLE">x</id>
                <apply>
                  <id xml:space="preserve" labels="*">LambertW</id>
                  <id xml:space="preserve" labels="VARIABLE">x</id>
                </apply>
              </apply>
              <id xml:space="preserve" labels="VARIABLE">x</id>
            </apply>
          </parens>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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            <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">LambertW</id>
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            <id xml:space="preserve" labels="*">LambertW</id>
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            <plus />
            <apply>
              <mult />
              <id xml:space="preserve" labels="VARIABLE">x</id>
              <apply>
                <id xml:space="preserve" labels="*">LambertW</id>
                <id xml:space="preserve" labels="VARIABLE">x</id>
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            <id xml:space="preserve" labels="VARIABLE">x</id>
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                    <real>2</real>
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                    <real>3</real>
                    <real>2</real>
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                  <real>8</real>
                  <real>3</real>
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                  <real>4</real>
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                <real>125</real>
                <real>24</real>
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                <pow />
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                <real>5</real>
              </apply>
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            <apply>
              <div />
              <real>54</real>
              <real>5</real>
            </apply>
            <apply>
              <pow />
              <id xml:space="preserve" labels="*">x</id>
              <real>6</real>
            </apply>
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