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cot(z)=-3*i/5 la ecuación

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Solución numérica:

Buscar la solución numérica en el intervalo [, ]

Solución

Ha introducido [src]
         -3*I
cot(z) = ----
          5  
$$\cot{\left(z \right)} = \frac{\left(-1\right) 3 i}{5}$$
Solución detallada
Tenemos la ecuación
$$\cot{\left(z \right)} = \frac{\left(-1\right) 3 i}{5}$$
cambiamos
$$\cot{\left(z \right)} - 1 + \frac{3 i}{5} = 0$$
$$\cot{\left(z \right)} - 1 - \frac{\left(-1\right) 3 i}{5} = 0$$
Sustituimos
$$w = \cot{\left(z \right)}$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
-1 + w - -3*i/5 = 0

Sumamos los términos semejantes en el miembro izquierdo de la ecuación:
-1 + w + 3*i/5 = 0

Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w + \frac{3 i}{5} = 1$$
Move the summands with the other variables
del miembro izquierdo al derecho, obtenemos:
$$\frac{\left(-3\right) i}{5} + w + \frac{3 i}{5} = \frac{\left(-3\right) i}{5} + 1$$
Dividamos ambos miembros de la ecuación en (w - 3*i/5 + 3*i/5)/w
w = 1 - 3*i/5 / ((w - 3*i/5 + 3*i/5)/w)

Obtenemos la respuesta: w = 1 - 3*i/5
hacemos cambio inverso
$$\cot{\left(z \right)} = w$$
sustituimos w:
Gráfica
Suma y producto de raíces [src]
suma
-im(acoth(3/5)) + I*re(acoth(3/5))
$$- \operatorname{im}{\left(\operatorname{acoth}{\left(\frac{3}{5} \right)}\right)} + i \operatorname{re}{\left(\operatorname{acoth}{\left(\frac{3}{5} \right)}\right)}$$
=
-im(acoth(3/5)) + I*re(acoth(3/5))
$$- \operatorname{im}{\left(\operatorname{acoth}{\left(\frac{3}{5} \right)}\right)} + i \operatorname{re}{\left(\operatorname{acoth}{\left(\frac{3}{5} \right)}\right)}$$
producto
-im(acoth(3/5)) + I*re(acoth(3/5))
$$- \operatorname{im}{\left(\operatorname{acoth}{\left(\frac{3}{5} \right)}\right)} + i \operatorname{re}{\left(\operatorname{acoth}{\left(\frac{3}{5} \right)}\right)}$$
=
-im(acoth(3/5)) + I*re(acoth(3/5))
$$- \operatorname{im}{\left(\operatorname{acoth}{\left(\frac{3}{5} \right)}\right)} + i \operatorname{re}{\left(\operatorname{acoth}{\left(\frac{3}{5} \right)}\right)}$$
-im(acoth(3/5)) + i*re(acoth(3/5))
Respuesta rápida [src]
z1 = -im(acoth(3/5)) + I*re(acoth(3/5))
$$z_{1} = - \operatorname{im}{\left(\operatorname{acoth}{\left(\frac{3}{5} \right)}\right)} + i \operatorname{re}{\left(\operatorname{acoth}{\left(\frac{3}{5} \right)}\right)}$$
z1 = -im(acoth(3/5)) + i*re(acoth(3/5))
Respuesta numérica [src]
z1 = -95.8185759344887 + 0.693147180559945*i
z2 = -42.4115008234622 + 0.693147180559945*i
z3 = 20.4203522483337 + 0.693147180559945*i
z4 = -36.1283155162826 + 0.693147180559945*i
z5 = -14.1371669411541 + 0.693147180559945*i
z6 = 67.5442420521806 + 0.693147180559945*i
z7 = -64.4026493985908 + 0.693147180559945*i
z8 = -89.5353906273091 + 0.693147180559945*i
z9 = 17.2787595947439 + 0.693147180559945*i
z10 = -80.1106126665397 + 0.693147180559945*i
z11 = 36.1283155162826 + 0.693147180559945*i
z12 = 61.261056745001 + 0.693147180559945*i
z13 = -32.9867228626928 + 0.693147180559945*i
z14 = -86.3937979737193 + 0.693147180559945*i
z15 = 54.9778714378214 + 0.693147180559945*i
z16 = 14.1371669411541 + 0.693147180559945*i
z17 = -54.9778714378214 + 0.693147180559945*i
z18 = -23.5619449019235 + 0.693147180559945*i
z19 = 80.1106126665397 + 0.693147180559945*i
z20 = 4.71238898038469 + 0.693147180559945*i
z21 = -1.5707963267949 + 0.693147180559945*i
z22 = 29.845130209103 + 0.693147180559945*i
z23 = 76.9690200129499 + 0.693147180559945*i
z24 = -51.8362787842316 + 0.693147180559945*i
z25 = -29.845130209103 + 0.693147180559945*i
z26 = 95.8185759344887 + 0.693147180559945*i
z27 = -10.9955742875643 + 0.693147180559945*i
z28 = 23.5619449019235 + 0.693147180559945*i
z29 = 1.5707963267949 + 0.693147180559945*i
z30 = 73.8274273593601 + 0.693147180559945*i
z31 = 98.9601685880785 + 0.693147180559945*i
z32 = 39.2699081698724 + 0.693147180559945*i
z33 = -48.6946861306418 + 0.693147180559945*i
z34 = -67.5442420521806 + 0.693147180559945*i
z35 = -73.8274273593601 + 0.693147180559945*i
z36 = 86.3937979737193 + 0.693147180559945*i
z37 = -76.9690200129499 + 0.693147180559945*i
z38 = 48.6946861306418 + 0.693147180559945*i
z39 = 64.4026493985908 + 0.693147180559945*i
z40 = 42.4115008234622 + 0.693147180559945*i
z41 = -83.2522053201295 + 0.693147180559945*i
z42 = -98.9601685880785 + 0.693147180559945*i
z43 = 45.553093477052 + 0.693147180559945*i
z44 = -45.553093477052 + 0.693147180559945*i
z45 = -7.85398163397448 + 0.693147180559945*i
z46 = -4.71238898038469 + 0.693147180559945*i
z47 = 89.5353906273091 + 0.693147180559945*i
z48 = 7.85398163397448 + 0.693147180559945*i
z49 = 92.6769832808989 + 0.693147180559945*i
z50 = -26.7035375555132 + 0.693147180559945*i
z51 = 70.6858347057703 + 0.693147180559945*i
z52 = 26.7035375555132 + 0.693147180559945*i
z53 = 83.2522053201295 + 0.693147180559945*i
z54 = -92.6769832808989 + 0.693147180559945*i
z55 = 10.9955742875643 + 0.693147180559945*i
z56 = -39.2699081698724 + 0.693147180559945*i
z57 = 32.9867228626928 + 0.693147180559945*i
z58 = -20.4203522483337 + 0.693147180559945*i
z59 = 58.1194640914112 + 0.693147180559945*i
z60 = 51.8362787842316 + 0.693147180559945*i
z61 = -70.6858347057703 + 0.693147180559945*i
z62 = -61.261056745001 + 0.693147180559945*i
z63 = -58.1194640914112 + 0.693147180559945*i
z64 = -17.2787595947439 + 0.693147180559945*i
z64 = -17.2787595947439 + 0.693147180559945*i