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Simplificación de expresiones/ Descomposición en fracciones simples/ log(z+1)/2-log(z)+log(z-1)/2+1/(2*z^2)

¿Cómo vas a descomponer esta log(z+1)/2-log(z)+log(z-1)/2+1/(2*z^2) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
log(z + 1)            log(z - 1)    1  
---------- - log(z) + ---------- + ----
    2                     2           2
                                   2*z
$$\left(\left(- \log{\left(z \right)} + \frac{\log{\left(z + 1 \right)}}{2}\right) + \frac{\log{\left(z - 1 \right)}}{2}\right) + \frac{1}{2 z^{2}}$$ 
log(z + 1)/2 - log(z) + log(z - 1)/2 + 1/(2*z^2)
Simplificación general [src] 
 1     log(1 + z)   log(-1 + z)         
---- + ---------- + ----------- - log(z)
   2       2             2              
2*z
$$- \log{\left(z \right)} + \frac{\log{\left(z - 1 \right)}}{2} + \frac{\log{\left(z + 1 \right)}}{2} + \frac{1}{2 z^{2}}$$ 
1/(2*z^2) + log(1 + z)/2 + log(-1 + z)/2 - log(z)
Respuesta numérica [src] 
-log(z) + 0.5/z^2 + 0.5*log(z + 1) + 0.5*log(z - 1)
-log(z) + 0.5/z^2 + 0.5*log(z + 1) + 0.5*log(z - 1)
Parte trigonométrica [src] 
 1     log(1 + z)   log(-1 + z)         
---- + ---------- + ----------- - log(z)
   2       2             2              
2*z
$$- \log{\left(z \right)} + \frac{\log{\left(z - 1 \right)}}{2} + \frac{\log{\left(z + 1 \right)}}{2} + \frac{1}{2 z^{2}}$$ 
1/(2*z^2) + log(1 + z)/2 + log(-1 + z)/2 - log(z)
Denominador común [src] 
 1     log(1 + z)   log(-1 + z)         
---- + ---------- + ----------- - log(z)
   2       2             2              
2*z
$$- \log{\left(z \right)} + \frac{\log{\left(z - 1 \right)}}{2} + \frac{\log{\left(z + 1 \right)}}{2} + \frac{1}{2 z^{2}}$$ 
1/(2*z^2) + log(1 + z)/2 + log(-1 + z)/2 - log(z)
Denominador racional [src] 
       2             2                 2            
2 - 4*z *log(z) + 2*z *log(1 + z) + 2*z *log(-1 + z)
----------------------------------------------------
                           2                        
                        4*z
$$\frac{- 4 z^{2} \log{\left(z \right)} + 2 z^{2} \log{\left(z - 1 \right)} + 2 z^{2} \log{\left(z + 1 \right)} + 2}{4 z^{2}}$$ 
(2 - 4*z^2*log(z) + 2*z^2*log(1 + z) + 2*z^2*log(-1 + z))/(4*z^2)
Combinatoria [src] 
     2               2                  2       
1 + z *log(1 + z) + z *log(-1 + z) - 2*z *log(z)
------------------------------------------------
                         2                      
                      2*z
$$\frac{- 2 z^{2} \log{\left(z \right)} + z^{2} \log{\left(z - 1 \right)} + z^{2} \log{\left(z + 1 \right)} + 1}{2 z^{2}}$$ 
(1 + z^2*log(1 + z) + z^2*log(-1 + z) - 2*z^2*log(z))/(2*z^2)
Potencias [src] 
 1     log(1 + z)   log(-1 + z)         
---- + ---------- + ----------- - log(z)
   2       2             2              
2*z
$$- \log{\left(z \right)} + \frac{\log{\left(z - 1 \right)}}{2} + \frac{\log{\left(z + 1 \right)}}{2} + \frac{1}{2 z^{2}}$$ 
1/(2*z^2) + log(1 + z)/2 + log(-1 + z)/2 - log(z)
Abrimos la expresión [src] 
 1              log(z + 1)   log(z - 1)
---- - log(z) + ---------- + ----------
   2                2            2     
2*z
$$- \log{\left(z \right)} + \frac{\log{\left(z - 1 \right)}}{2} + \frac{\log{\left(z + 1 \right)}}{2} + \frac{1}{2 z^{2}}$$ 
1/(2*z^2) - log(z) + log(z + 1)/2 + log(z - 1)/2
Compilar la expresión [src] 
 1     log(z + 1)   log(z - 1)         
---- + ---------- + ---------- - log(z)
   2       2            2              
2*z
$$- \log{\left(z \right)} + \frac{\log{\left(z - 1 \right)}}{2} + \frac{\log{\left(z + 1 \right)}}{2} + \frac{1}{2 z^{2}}$$ 
1/(2*z^2) + log(z + 1)/2 + log(z - 1)/2 - log(z)
Unión de expresiones racionales [src] 
     2                                       
1 + z *(-2*log(z) + log(1 + z) + log(-1 + z))
---------------------------------------------
                        2                    
                     2*z
$$\frac{z^{2} \left(- 2 \log{\left(z \right)} + \log{\left(z - 1 \right)} + \log{\left(z + 1 \right)}\right) + 1}{2 z^{2}}$$ 
(1 + z^2*(-2*log(z) + log(1 + z) + log(-1 + z)))/(2*z^2)
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