Esempio: funzioni iperboliche e iperboliche inverse

Funzioni > Trigonometriche, logaritmiche ed esponenziali > Esempio: funzioni iperboliche e iperboliche inverse

Mostrare che il passaggio del risultato della valutazione di una funzione iperbolica a una funzione iperbolica inversa corrispondente restituisce l'argomento di input originale.

1. Definire e valutare l'argomento di input z simbolicamente e numericamente:

2. Valutare sinh e passare il risultato a asinh e quindi valutarlo simbolicamente e numericamente:

<region id="ID0E5CBEB" actualWidth="259.17666666666668" actualHeight="51.158666666666669" top="172.8" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="17">
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <symEval>
        <apply>
          <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">asinh</id>
          <apply>
            <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">sinh</id>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">z</id>
          </apply>
        </apply>
        <command>
          <placeholder />
        </command>
        <symResult>
          <apply>
            <plus />
            <apply>
              <div />
              <id labels="CONSTANT" xml:space="preserve">π</id>
              <real>2</real>
            </apply>
            <imag symbol="i">1</imag>
          </apply>
        </symResult>
      </symEval>
      <unitOverride>
        <placeholder />
      </unitOverride>
    </eval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

Come previsto, i risultati simbolico e numerico sono entrambi uguali al valore di z.

3. Ripetere il passo precedente utilizzando le funzioni cosh e acosh:

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  <math resultRef="19">
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <symEval>
        <apply>
          <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">acosh</id>
          <apply>
            <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">cosh</id>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">z</id>
          </apply>
        </apply>
        <command>
          <placeholder />
        </command>
        <symResult>
          <apply>
            <plus />
            <apply>
              <div />
              <id labels="CONSTANT" xml:space="preserve">π</id>
              <real>2</real>
            </apply>
            <imag symbol="i">1</imag>
          </apply>
        </symResult>
      </symEval>
      <unitOverride>
        <placeholder />
      </unitOverride>
    </eval>
  </math>
</region>

4. Ripetere il passo precedente utilizzando le funzioni tanh e atanh:

<region id="ID0EFJBEB" actualWidth="266.21666666666664" actualHeight="51.158666666666669" top="364.80000000000007" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="21">
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <symEval>
        <apply>
          <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">atanh</id>
          <apply>
            <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">tanh</id>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">z</id>
          </apply>
        </apply>
        <command>
          <placeholder />
        </command>
        <symResult>
          <apply>
            <plus />
            <apply>
              <div />
              <id labels="CONSTANT" xml:space="preserve">π</id>
              <real>2</real>
            </apply>
            <imag symbol="i">1</imag>
          </apply>
        </symResult>
      </symEval>
      <unitOverride>
        <placeholder />
      </unitOverride>
    </eval>
  </math>
</region>

5. Ripetere il passo precedente utilizzando le funzioni coth e acoth:

<region id="ID0EEMBEB" actualWidth="263.72333333333336" actualHeight="51.158666666666669" top="460.80000000000007" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="23">
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <symEval>
        <apply>
          <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">acoth</id>
          <apply>
            <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">coth</id>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">z</id>
          </apply>
        </apply>
        <command>
          <placeholder />
        </command>
        <symResult>
          <apply>
            <plus />
            <apply>
              <div />
              <id labels="CONSTANT" xml:space="preserve">π</id>
              <real>2</real>
            </apply>
            <imag symbol="i">1</imag>
          </apply>
        </symResult>
      </symEval>
      <unitOverride>
        <placeholder />
      </unitOverride>
    </eval>
  </math>
</region>

6. Ripetere il passo precedente utilizzando le funzioni sech e asech:

<region id="ID0EDPBEB" actualWidth="261.96333333333331" actualHeight="51.158666666666669" top="556.80000000000007" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="25">
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <symEval>
        <apply>
          <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">asech</id>
          <apply>
            <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">sech</id>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">z</id>
          </apply>
        </apply>
        <command>
          <placeholder />
        </command>
        <symResult>
          <apply>
            <plus />
            <apply>
              <div />
              <id labels="CONSTANT" xml:space="preserve">π</id>
              <real>2</real>
            </apply>
            <imag symbol="i">1</imag>
          </apply>
        </symResult>
      </symEval>
      <unitOverride>
        <placeholder />
      </unitOverride>
    </eval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

7. Ripetere il passo precedente utilizzando le funzioni csch e acsch:

<region id="ID0ECSBEB" actualWidth="261.96333333333331" actualHeight="51.158666666666669" top="652.80000000000007" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="27">
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <symEval>
        <apply>
          <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">acsch</id>
          <apply>
            <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">csch</id>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">z</id>
          </apply>
        </apply>
        <command>
          <placeholder />
        </command>
        <symResult>
          <apply>
            <plus />
            <apply>
              <div />
              <id labels="CONSTANT" xml:space="preserve">π</id>
              <real>2</real>
            </apply>
            <imag symbol="i">1</imag>
          </apply>
        </symResult>
      </symEval>
      <unitOverride>
        <placeholder />
      </unitOverride>
    </eval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

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Funzioni iperboliche

Funzioni iperboliche inverse

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