예제: 디락 델타(단위 충격) 및 LambertW 함수

함수 > 기호 함수 > 예제: 디락 델타(단위 충격) 및 LambertW 함수

이 예제에서는 디락 델타 함수와 LambertW 함수 속성에 대해 설명합니다.

디락 델타 함수

디락 델타 함수 delta에는 다음과 같은 속성이 있습니다.

다음 변환은 디락 델타 함수로 결과를 반환합니다.

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      <apply>
        <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">Φ</id>
        <id xml:space="preserve">τ</id>
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          <id xml:space="preserve">fourier</id>
          <id xml:space="preserve">τ</id>
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          <div />
          <apply>
            <plus />
            <apply>
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                <id xml:space="preserve" labels="*">ω</id>
                <id xml:space="preserve" labels="CONSTANT">π</id>
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              <apply>
                <id xml:space="preserve" labels="FUNCTION">Δ</id>
                <id xml:space="preserve" labels="*">ω</id>
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            </apply>
            <imag symbol="i">1</imag>
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          <id xml:space="preserve" labels="*">ω</id>
        </apply>
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위 예에서 Φ는 헤비사이드 계단 함수입니다.

LambertW 함수

다음과 같은 방정식의 기호 해에 LambertW 함수가 나타납니다.

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        <real>0</real>
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          <neg />
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                <real>5</real>
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            <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">e</id>
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              <neg />
              <id xml:space="preserve">v</id>
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          <id xml:space="preserve">v</id>
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          <apply>
            <minus />
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              <mult />
              <id xml:space="preserve" labels="*">b</id>
              <apply>
                <id xml:space="preserve" labels="*">LambertW</id>
                <apply>
                  <div />
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                    <mult />
                    <id xml:space="preserve" labels="*">c</id>
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                      <id xml:space="preserve" labels="CONSTANT">e</id>
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                        <id xml:space="preserve" labels="*">a</id>
                        <id xml:space="preserve" labels="*">b</id>
                      </apply>
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                  <id xml:space="preserve" labels="*">b</id>
                </apply>
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            </apply>
            <id xml:space="preserve" labels="*">a</id>
          </apply>
          <id xml:space="preserve" labels="*">b</id>
        </apply>
      </symResult>
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        <apply>
          <id xml:space="preserve" labels="*">LambertW</id>
          <apply>
            <div />
            <real>15</real>
            <real>2</real>
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      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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다음은 LambertW 함수의 적분, 도함수 및 테일러 급수 전개입니다.

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              <id xml:space="preserve" labels="*">LambertW</id>
              <id xml:space="preserve" labels="VARIABLE">x</id>
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            <apply>
              <minus />
              <apply>
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                <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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