2^x=x (уравнение)

Найду корень уравнения: 2^x=x

Вы ввели

 x    
2  = x

$$2^{x} = x$$

Быстрый ответ

       re(W(-log(2)))   I*im(W(-log(2)))
x1 = - -------------- - ----------------
           log(2)            log(2)     

$$x_{1} = - \frac{\operatorname{re}{\left(W\left(- \log{\left(2 \right)}\right)\right)}}{\log{\left(2 \right)}} - \frac{i \operatorname{im}{\left(W\left(- \log{\left(2 \right)}\right)\right)}}{\log{\left(2 \right)}}$$

Сумма и произведение корней

сумма

      re(W(-log(2)))   I*im(W(-log(2)))
0 + - -------------- - ----------------
          log(2)            log(2)     

$$0 - \left(\frac{\operatorname{re}{\left(W\left(- \log{\left(2 \right)}\right)\right)}}{\log{\left(2 \right)}} + \frac{i \operatorname{im}{\left(W\left(- \log{\left(2 \right)}\right)\right)}}{\log{\left(2 \right)}}\right)$$

=

  re(W(-log(2)))   I*im(W(-log(2)))
- -------------- - ----------------
      log(2)            log(2)     

$$- \frac{\operatorname{re}{\left(W\left(- \log{\left(2 \right)}\right)\right)}}{\log{\left(2 \right)}} - \frac{i \operatorname{im}{\left(W\left(- \log{\left(2 \right)}\right)\right)}}{\log{\left(2 \right)}}$$

произведение

  /  re(W(-log(2)))   I*im(W(-log(2)))\
1*|- -------------- - ----------------|
  \      log(2)            log(2)     /

$$1 \left(- \frac{\operatorname{re}{\left(W\left(- \log{\left(2 \right)}\right)\right)}}{\log{\left(2 \right)}} - \frac{i \operatorname{im}{\left(W\left(- \log{\left(2 \right)}\right)\right)}}{\log{\left(2 \right)}}\right)$$

=

-re(W(-log(2))) - I*im(W(-log(2)))
----------------------------------
              log(2)              

$$\frac{- \operatorname{re}{\left(W\left(- \log{\left(2 \right)}\right)\right)} - i \operatorname{im}{\left(W\left(- \log{\left(2 \right)}\right)\right)}}{\log{\left(2 \right)}}$$

Численный ответ

x1 = 0.824678546142074 - 1.56743212384965*i

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