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Solución numérica:

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Solución

Ha introducido [src] 
        x    
pi*x - E  = 1
$$- e^{x} + \pi x = 1$$
Simplificar
Gráfica
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
            / /  1  \\       / /  1  \\
            | |  -- ||       | |  -- ||
            | |  pi ||       | |  pi ||
     1      | |-e   ||       | |-e   ||
x1 = -- - re|W|-----|| - I*im|W|-----||
     pi     \ \  pi //       \ \  pi //
$$x_{1} = \frac{1}{\pi} - \operatorname{re}{\left(W\left(- \frac{e^{\frac{1}{\pi}}}{\pi}\right)\right)} - i \operatorname{im}{\left(W\left(- \frac{e^{\frac{1}{\pi}}}{\pi}\right)\right)}$$ 
x1 = 1/pi - re(LambertW(-exp(1/pi)/pi)) - i*im(LambertW(-exp(1/pi)/pi))
Suma y producto de raíces [src] 
suma
       / /  1  \\       / /  1  \\
       | |  -- ||       | |  -- ||
       | |  pi ||       | |  pi ||
1      | |-e   ||       | |-e   ||
-- - re|W|-----|| - I*im|W|-----||
pi     \ \  pi //       \ \  pi //
$$\frac{1}{\pi} - \operatorname{re}{\left(W\left(- \frac{e^{\frac{1}{\pi}}}{\pi}\right)\right)} - i \operatorname{im}{\left(W\left(- \frac{e^{\frac{1}{\pi}}}{\pi}\right)\right)}$$ 
=
       / /  1  \\       / /  1  \\
       | |  -- ||       | |  -- ||
       | |  pi ||       | |  pi ||
1      | |-e   ||       | |-e   ||
-- - re|W|-----|| - I*im|W|-----||
pi     \ \  pi //       \ \  pi //
$$\frac{1}{\pi} - \operatorname{re}{\left(W\left(- \frac{e^{\frac{1}{\pi}}}{\pi}\right)\right)} - i \operatorname{im}{\left(W\left(- \frac{e^{\frac{1}{\pi}}}{\pi}\right)\right)}$$ 
producto
       / /  1  \\       / /  1  \\
       | |  -- ||       | |  -- ||
       | |  pi ||       | |  pi ||
1      | |-e   ||       | |-e   ||
-- - re|W|-----|| - I*im|W|-----||
pi     \ \  pi //       \ \  pi //
$$\frac{1}{\pi} - \operatorname{re}{\left(W\left(- \frac{e^{\frac{1}{\pi}}}{\pi}\right)\right)} - i \operatorname{im}{\left(W\left(- \frac{e^{\frac{1}{\pi}}}{\pi}\right)\right)}$$ 
=
       / /  1  \\       / /  1  \\
       | |  -- ||       | |  -- ||
       | |  pi ||       | |  pi ||
1      | |-e   ||       | |-e   ||
-- - re|W|-----|| - I*im|W|-----||
pi     \ \  pi //       \ \  pi //
$$\frac{1}{\pi} - \operatorname{re}{\left(W\left(- \frac{e^{\frac{1}{\pi}}}{\pi}\right)\right)} - i \operatorname{im}{\left(W\left(- \frac{e^{\frac{1}{\pi}}}{\pi}\right)\right)}$$ 
1/pi - re(LambertW(-exp(1/pi)/pi)) - i*im(LambertW(-exp(1/pi)/pi))
Respuesta numérica [src] 
x1 = 1.20214112769942 - 0.583538037274717*i
x1 = 1.20214112769942 - 0.583538037274717*i
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