Álgebra-lineal-II: diagonalizar-y-canonizar-y-triangulizar-matrices y clausula-y-condenación-de-este-mundo y análisis-funcional

Diagonalizar una matriz:

F(x,y) = ( < a,b >,< c,d > ) o < x,y >

Valores propios:

det( F+(-1)·Id(x,x) ) = | < a+(-x),b >,< c,d+(-x) > | = (a+(-x))·(d+(-x))+(-1)·cb = 0

x_{1} = (1/2)·( (a+b)+( (a+b)^{2}+(-4)·(ab+(-1)·cd) )^{(1/2)} ) = p

x_{2} = (1/2)·( (a+b)+(-1)·( (a+b)^{2}+(-4)·(ab+(-1)·cd) )^{(1/2)} ) = q

det( F+(-1)·Id(p,p) ) = 0

det( F+(-1)·Id(q,q) ) = 0

Vectores propios:

( F+(-1)·Id(p,p) ) o < x,y > = 0

( F+(-1)·Id(q,q) ) o < x,y > = 0

Diagonalización:

Y o F o X = Id(p,q)

F(x,y) = ( < a,a >,< a,a > ) o < x,y >

Valores propios:

c = 0 & d = 2a

Vectores propios:

u = < s,(-s) > & v = < s,s >

Impuestos socialistas:

p = |< s,(-s) >| = |< s,s >| = (1.41)·s

Diagonalización:

F o X = ( < a,a >,< a,a > ) o ( < s,s >,< (-s),s > ) = ( < 0,2as >,< 0,2as > )

Y o F o X = ( 1/(2s^{2}) )·( < s,(-s) >,< s,s > ) o ( < 0,2as >,< 0,2as > ) = ( < 0,0 >,< 0,2a > )

F(x,y) = ( < a,ai >,< (-a)·i,a > ) o < x,y >

Valores propios:

c = 0 & d = 2a

Vectores propios:

u = < si,(-s) > & v = < (-s),si >

Impuestos social-demócratas:

p = |< si,(-s) >| = |< (-s),si >| = 0

Diagonalización:

F o X = ( < a,ai >,< (-a)·i,a > ) o ( < si,(-s) >,< (-s),si > ) = ( < 0,(-2)·as >,< 0,2asi > )

Y o F o X = ( 1/((-2)·s^{2}) )·( < si,s >,< s,si > ) o ( < 0,(-2)·as >,< 0,2asi > ) = ( < 0,0 >,< 0,2a > )

F(x,y) = ( < a,b >,< b,a > ) o < x,y >

Valores propios:

c = a+(-b) & d = a+b

Vectores propios:

u = < s,(-s) > & v = < s,s >

Impuestos socialistas:

p = |< s,(-s) >| = |< s,s >| = (1.41)·s

Diagonalización:

F o X = ( < a,b >,< b,a > ) o ( < s,s >,< (-s),s > ) = ( < (a+(-b))·s,(a+b)·s >,< (b+(-a))·s,(b+a)·s > )

Y o F o X = ( 1/(2s^{2}) )·( < s,(-s) >,< s,s > ) o ( < (a+(-b))·s,(a+b)·s >,< (b+(-a))·s,(b+a)·s > ) = ...

... ( < a+(-b),0 >,< 0,a+b > )

F(x,y) = ( < a,b >,< (-b),a > ) o < x,y >

Valores propios:

c = a+(-b)·i & d = a+bi

Vectores propios:

u = < si,s > & v = < s,si >

Impuestos social-demócratas:

p = |< si,s >| = |< s,si >| = 0

Diagonalización:

F o X = ( < a,b >,< (-b),a > ) o ( < si,s >,< s,si > ) = ( < (ai+b)·s,(a+bi)·s >,< ((-b)·i+a)·s,((-b)+ai)·s > )

Y o F o X = ...

... ( 1/((-2)·s^{2}) )·( < si,(-s) >,< (-s),si > ) o ( < (ai+b)·s,(a+bi)·s >,< ((-b)·i+a)·s,((-b)+ai)·s > ) = ...

... ( < a+(-b)·i,0 >,< 0,a+bi > )

Canonizar una matriz:

F(x,y) = ( < a,b >,< c,d > ) o < x,y >

Valores propios canónicos:

det( F+(-1)·Id(x,y) ) = | < a+(-x),b >,< c,d+(-y) > | = 0

x = a+bi = p & y = d+(-c)·i = q

det( F+(-1)·Id(y,x) ) = | < a+(-y),b >,< c,d+(-x) > | = 0

x = d+bi = p & y = a+(-c)·i = q

Vectores propios canónicos:

( F+(-1)·Id(p,q) ) o < x,y > = 0

( F+(-1)·Id(q,p) ) o < x,y > = 0

Canonización:

Y o F o X = F

F o X = X o F

F(x,y) = ( < a,a >,< a,a > ) o < x,y >

Valores propios canónicos:

c = a+(-a)·i & d = a+ai

Vectores propios canónicos:

u = < si,s > & v = < s,si >

Impuestos social-demócratas:

p = |< si,s >| = |< s,si >| = 0

Canonización:

F o X = ( < a,a >,< a,a > ) o ( < si,s >,< s,si > ) = ( < (ai+a)·s,(ai+a)·s >,< (a+ai)·s,(a+ai)·s > )

Y o F o X = ...

... ( 1/((-2)·s^{2}) )·( < si,(-s) >,< (-s),si > ) o ( < (ai+a)·s,(ai+a)·s >,< (a+ai)·s,(a+ai)·s > ) = ...

... ( < a,a >,< a,a > )

X o F = ( < si,s >,< s,si > ) o ( < a,a >,< a,a > ) = ( < (ai+a)·s,(a+ai)·s >,< (ai+a)·s,(a+ai)·s > )

F(x,y) = ( < a,ai >,< (-a)·i,a > ) o < x,y >

Valores propios canónicos:

c = a+(-a)·i & d = a+ai

Vectores propios canónicos:

u = < s,(-s) > & v = < s,s >

Impuestos socialistas:

p = |< s,(-s) >| = |< s,s >| = (1.41)·s

Canonización:

F o X = ( < a,ai >,< (-a)·i,a > ) o ( < s,s >,< (-s),s > ) = ...

... ( < (a+(-a)·i)·s,(a+ai)·s >,< ((-a)·i+(-a))·s,((-a)·i+a)·s > )

Y o F o X = ...

... ( 1/(2s^{2}) )·( < s,(-s) >,< s,s > ) o ( < (a+(-a)·i)·s,(a+ai)·s >,< ((-a)·i+(-a))·s,((-a)·i+a)·s > ) = ...

... ( < a,ai >,< (-a)·i,a > )

X o F = ( < s,s >,< (-s),s > ) o ( < a,ai >,< (-a)·i,a > ) = ...

... ( < (a+(-a)·i)·s,(ai+a)·s >,< ((-a)+(-a)·i)·s,((-a)·i+a)·s > )

F(x,y) = ( < a,b >,< b,a > ) o < x,y >

Valores propios canónicos:

c = a+(-b)·i & d = a+bi

Vectores propios canónicos:

u = < si,s > & v = < s,si >

Impuestos social-demócratas:

p = |< si,s >| = |< s,si >| = 0

Canonización:

F o X = ( < a,b >,< b,a > ) o ( < si,s >,< s,si > ) = ( < (ai+b)·s,(bi+a)·s >,< (a+bi)·s,(b+ai)·s > )

Y o F o X = ...

... ( 1/((-2)·s^{2}) )·( < si,(-s) >,< (-s),si > ) o ( < (ai+b)·s,(bi+a)·s >,< (a+bi)·s,(b+ai)·s > ) = ...

... ( < a,b >,< b,a > )

X o F = ( < si,s >,< s,si > ) o ( < a,b >,< b,a > ) = ( < (ai+b)·s,(a+bi)·s >,< (bi+a)·s,(b+ai)·s > )

F(x,y) = ( < a,b >,< (-b),a > ) o < x,y >

Valores propios canónicos:

c = a+(-b) & d = a+b

Vectores propios canónicos:

u = < s,(-s) > & v = < s,s >

Impuestos socialistas:

p = |< s,(-s) >| = |< s,s >| = (1.41)·s

Canonización:

F o X = ( < a,b >,< (-b),a > ) o ( < s,s >,< (-s),s > ) = ...

... ( < (a+(-b))·s,(a+b)·s >,< ((-b)+(-a))·s,((-b)+a)·s > )

Y o F o X = ...

... ( 1/(2s^{2}) )·( < s,(-s) >,< s,s > ) o ( < (a+(-b))·s,(a+b)·s >,< ((-b)+(-a))·s,((-b)+a)·s > ) = ...

... ( < a,b >,< (-b),a > )

X o F = ( < s,s >,< (-s),s > ) o ( < a,b >,< (-b),a > ) = ...

... ( < (a+(-b))·s,(b+a)·s >,< ((-a)+(-b))·s,((-b)+a)·s > )

Examen 1 de Álgebra lineal II

Diagonalizad y Canonizad el siguiente endomorfismo:

F(x,y) = ( < 1,1 >,< 1,1 > ) o < x,y >

Utilizad una base de vectores propios numérica.

Examen 2 de Álgebra lineal II

Diagonalizad y Canonizad el siguiente endomorfismo:

F(x,y) = ( < 1,i >,< (-i),1 > ) o < x,y >

Utilizad una base de vectores propios numérica.

Ley: [ de 2 osciladores paralelos y de 2 osciladores ortogonales paralelos ]

m·d_{tt}^{2}[z(t)] = (-1)·( < k,k >,< k,k > )·z(t)

z(t) = e^{( X o ( < 0,0 >,< 0,2·(k/m) > ) o Y )^{(1/2)}·it}

z(t) = e^{( X o ( < 0,0 >,< 0,2·(k/m) > ) o Y )^{(1/2)}·(-1)·it}

Motor de cuatro tiempos con cuatro pistones exteriores.

El interior del sistema gira.

m·d_{tt}^{2}[z(t)] = (-i)·( < k,k >,< k,k > )·z(t)

z(t) = e^{( X o ( < 0,0 >,< 0,2·(k/m) > ) o Y )^{(1/2)}·(1/2)^{(1/2)}·(1+(-i))·t}

z(t) = e^{( X o ( < 0,0 >,< 0,2·(k/m) > ) o Y )^{(1/2)}·(-1)·(1/2)^{(1/2)}·(1+(-i))·t}

Motor de cuatro tiempos de prótesis del cuerpo.

El exterior del sistema gira.

Ley: [ de 2 osciladores paralelos y de 2 pesos ortogonales paralelos ]

m·d_{tt}^{2}[z(t)] = (-1)·( < k,q·(g/l) >,< q·(g/l),k > )·z(t)

z(t) = e^{( X o ( < (k/m)+(-1)·(q/m)·(g/l),0 >,< 0,(k/m)+(q/m)·(g/l) > ) o Y )^{(1/2)}·it}

z(t) = e^{( X o ( < (k/m)+(-1)·(q/m)·(g/l),0 >,< 0,(k/m)+(q/m)·(g/l) > ) o Y )^{(1/2)}·(-1)·it}

m·d_{tt}^{2}[z(t)] = (-1)·( < k,(-q)·(g/l) >,< (-q)·(g/l),k > )·z(t)

z(t) = e^{( X o ( < (k/m)+(-1)·(q/m)·(g/l),0 >,< 0,(k/m)+(q/m)·(g/l) > ) o Y )^{(1/2)}·it}

z(t) = e^{( X o ( < (k/m)+(-1)·(q/m)·(g/l),0 >,< 0,(k/m)+(q/m)·(g/l) > ) o Y )^{(1/2)}·(-1)·it}

Motor de dos tiempos carburo-eléctrico con dos pistones exteriores.

Triangulización en cajas de Jordan:

Valores propios:

( z = p & w = q ) || ( z = q & w = p )

Ecuación característica:

[EG][Ez][ det(G) = 0 & z es valor propio & G o ( F+(-1)·Id(z,z) ) = 0 ]

Vector propio de la ecuación característica:

( F+(-1)·Id(z,z) ) o < x,y > = 0

Triangulización en caja de Jordan:

< u,v > es el vector de Jordan.

X = ( < x,u >,< y,v > )

Y o F o X = Id(z,w)+( < 0,1 >,< 0,0 > )

F(x,y) = ( < a,a >,< a,a > ) o < x,y >

Valores propios:

c = 2a & d = 0 

Ecuación característica:

G o ( F+(-1)·Id(2a) ) = ( < 1,1 >,< 1,1 > ) o ( < (-a),a >,< a,(-a) > ) = 0

Vector propio de la ecuación característica:

u = < s,s >

Vector de Jordan:

v = s·< ( ((1/a)+(-1))/2 ),( ((1/a)+(-1))/2 )+1 >

F o X = ( < a,a >,< a,a > ) o ( < s,xs >,< s,ys > ) = ( < 2as,(ax+ay)·s >,< 2as,(ax+ay)·s > )

Y o F o X = ...

... ( 1/( (y+(-x))·s^{2} ) )·( < ys,(-1)·xs >,< (-s),s > ) o ( < 2as,(ax+ay)·s >,< 2as,(ax+ay)·s > ) = ...

... ( 1/( (y+(-x))·s^{2} ) )·( < 2as^{2}·(y+(-x)),as^{2}·( y^{2}+(-1)·x^{2} ) >,< 0,0 > ) = ...

... ( < 2a,1 >,< 0,0 > )

Si y = x+1 ==> y^{2}+(-1)·x^{2} = (x+1)^{2}+(-1)·x^{2} = 2x+1 = (1/a)

F(x,y) = ( < a,(-a) >,< (-a),a > ) o < x,y >

Valores propios:

c = 2a & d = 0 

Ecuación característica:

G o ( F+(-1)·Id(0) ) = ( < 1,1 >,< 1,1 > ) o ( < a,(-a) >,< (-a),a > ) = 0

Vector propio de la ecuación característica:

u = < s,s >

Vector de Jordan:

v = s·< (-1)·( ((1/a)+1)/2 ),(-1)·( ((1/a)+1)/2 )+1 >

F o X = ( < a,(-a) >,< (-a),a > ) o ( < s,xs >,< s,ys > ) = ( < 0,(ax+(-a)·y)·s >,< 0,((-a)·x+ay)·s > )

Y o F o X = ...

... ( 1/( (y+(-x))·s^{2} ) )·( < ys,(-1)·xs >,< (-s),s > ) o ( < 0,(ax+(-a)·y)·s >,< 0,((-a)·x+ay)·s > ) = ...

... ( 1/( (y+(-x))·s^{2} ) )·( < 0,as^{2}·( x^{2}+(-1)·y^{2} ) >,< 0,2a·(y+(-x)) > ) = ...

... ( < 0,1 >,< 0,2a > )

Si y = x+1 ==> x^{2}+(-1)·y^{2} = x^{2}+(-1)·(x+1)^{2} = (-2)·x+(-1) = (1/a)

F(x,y) = ( < a,ai >,< (-a)·i,a > ) o < x,y >

Valores propios:

c = 2a & d = 0 

Ecuación característica:

G o ( F+(-1)·Id(2a) ) = ( < 1,i >,< (-i),1 > ) o ( < (-a),ai >,< (-a)·i,(-a) > ) = 0

Vector propio de la ecuación característica:

u = < si,s >

Vector de Jordan:

v = s·< (-1)·( ((1/ai)+1)/2 ),i·( ((1/ai)+1)/2 )+(-i) >

F o X = ...

... ( < a,ai >,< (-a)·i,a > ) o ( < si,xs >,< s,ys > ) = ( < 2ais,(ax+ayi)·s >,< 2as,((-a)·xi+ay)·s > )

Y o F o X = ...

... ( 1/( (yi+(-x))·s^{2} ) )·...

... ( < ys,(-1)·xs >,< (-s),si > ) o ( < 2ais,(ax+ayi)·s >,< 2as,((-a)·xi+ay)·s > ) = ...

... ( 1/( (yi+(-x))·s^{2} ) )·( < 2as^{2}·(yi+(-x)),ais^{2}·( y^{2}+x^{2} ) >,< 0,0 > ) = ...

... ( < 2a,1 >,< 0,0 > )

Si yi = x+1 ==> y^{2}+x^{2} = (-1)·(x+1)^{2}+x^{2} = (-2)·x+(-1) = (1/ai)

F(x,y) = ( < a,(-a)·i >,< ai,a > ) o < x,y >

Valores propios:

c = 2a & d = 0 

Ecuación característica:

G o ( F+(-1)·Id(0) ) = ( < 1,i >,< (-i),1 > ) o ( < a,(-a)·i >,< ai,a > ) = 0

Vector propio de la ecuación característica:

u = < si,s >

Vector de Jordan:

v = s·< ( ((1/ai)+(-1))/2 ),(-i)·( ((1/ai)+(-1))/2 )+(-i) >

F o X = ...

... ( < a,(-a)·i >,< ai,a > ) o ( < si,xs >,< s,ys > ) = ( < 0,(ax+(-a)·yi)·s >,< 0,(axi+ay)·s > )

Y o F o X = ...

... ( 1/( (yi+(-x))·s^{2} ) )·...

... ( < ys,(-1)·xs >,< (-s),si > ) o ( < 0,(ax+(-a)·yi)·s >,< 0,(axi+ay)·s > ) = ...

... ( 1/( (yi+(-x))·s^{2} ) )·( < 0,ais^{2}·( (-1)·y^{2}+(-1)·x^{2} ) >,< 0,2as^{2}(yi+(-x)) > ) = ...

... ( < 0,1 >,< 0,2a > )

Si yi = x+1 ==> (-1)·y^{2}+(-1)·x^{2} = (x+1)^{2}+(-1)·x^{2} = 2x+1 = (1/ai)

F(x,y) = ( < a,b >,< b,a > ) o < x,y >

Valores propios:

c = a+(-b) & d = a+b 

Ecuación catacterística:

G o ( F+(-1)·Id(a+b) ) = ( < 1,1 >,< 1,1 > ) o ( < (-b),b >,< b,(-b) > ) = 0

Vector propio de la ecuación característica:

u = < s,s >

F(x,y) = ( < a,b >,< (-b),a > ) o < x,y >

Valores propios:

c = a+(-b)·i & d = a+bi 

Ecuación característica:

G o ( F+(-1)·Id(a+(-b)·i) ) = ( < 1,i >,< (-i),1 > ) o ( < bi,b >,< (-b),bi > ) = 0

Vector propio de la ecuación característica:

u = < si,s >

Examen 1 de Álgebra lineal II

Triangulizad en forma de caja de Jordan el siguiente endomorfismo:

F(x,y) = ( < 1,1 >,< 1,1 > ) o < x,y >

Examen 2 de Álgebra lineal II

Triangulizad en forma de caja de Jordan el siguiente endomorfismo:

F(x,y) = ( < 1,i >,< (-i),1 > ) o < x,y >

Uzkatzen-ten-dut-zû-tek a la gentotzak,

parlatzi-ten-dut-zare-dut en Euskera-Bascotzok parlatzi-koak, 

amb les meuotzaks orelli-koaks.

Veurtu-ten-dut-zû-tek a la gentotzak,

escrivitzi-ten-dut-zare-dut en Euskera-Bascotzok parlatzi-koak, 

amb els meuotzoks ur-ulli-koaks.

Clásicos:

Danzar [o] Dançar [o] Danzijjarri

Lanzar [o] Llançar [o] Llanzijjarri

Oreja [o] Orella [o] Orelli-koak

Oveja [o] Ovella [o] Ovelli-koak

Si no hubiese la clausula,

no habría condenación.

Hay la clausula,

y hay condenación.

Si Dios me odia a mi, te odia a ti,

y se tiene condenación,

porque Dios no ha puesto la clausula.

Si Dios no me odia a mi, no te odia a ti

y no se tiene condenación,

porque Dios ha puesto la clausula.

Si uno se salta la Ley,

tiene condenación.

Si uno no se salta la Ley,

no tiene condenación.

Teorema Constructor:

( x [< y & x >] y ) <==> x = y

( ¬( x [< y ) || ¬( x >] y ) ) <==> x != y

Teorema Destructor:

( x [< y & x >] y ) <==> x != y

( ¬( x [< y ) || ¬( x >] y ) ) <==> x = y

Teorema Constructor:

Si ( x = y & y = z ) ==> x = z

Teorema Destructor:

Si ( x = y & y = z ) ==> x != z

Demostración:

Si ( x = y & y = z ) ==>

( x [< y & x >] y & y [< z & y >] z )

( x [< y & y [< z & x >] y & y >] z )

x [< z & x >] z

x = z

Si ( x = y & y = z ) ==>

( x [< y & x >] y & y [< z & y >] z )

( x [< y & y [< z & x >] y & y >] z )

¬( x [< z & x >] z )

¬( x [< z ) || ¬( x >] z)

x != z

Clásicos:

Hotir [o] Fotre [o] Fotetzi

Batir [o] Batre [o] Batetzi

Joter [o] Jotre [o] Jotetzi

Cometer [o] Cometre [o] Cometetzi

I stare-kate joted or foted,

in hete moment, 

of puted.

I not stare-kate joted nit foted,

in shete moment,

of not puted.

Análisis funcional:

Base dual de aplicaciones lineales de vector función:

A[f(x)+g(x)] = k

Base ortogonal de aplicaciones lineales de vector función:

B[f(x)+g(x)] = 0

Teorema:

Sea P[f(x)] = d_{x}[ f(0) ] ==> f(x) = (-k)·e^{x} es vector dual de g(x) = 2kx

Teorema:

Sea Q[f(x)] = d_{x}[ f(0) ] ==> f(x) = (-k)·sin(x) es vector dual de g(x) = 2kx

Teorema:

Sea A[f(x)] = int[x = 0]-[pi·i][ f(x) ]d[x] ==> ...

... f(x) = (-k)·e^{x} es vector dual de g(x) = (1/pi)^{2}·2kx

Teorema:

Sea B[f(x)] = int[x = 0]-[pi][ f(x) ]d[x] ==> ...

... f(x) = k·sin(x) es vector dual de g(x) = (-1)·(1/pi)^{2}·2kx

Análisis funcional:

Base dual de aplicaciones lineales de vector función de dos variables:

A[f(x,y)+g(x,y)] = k

Base ortogonal de aplicaciones lineales de vector función de dos variables:

B[f(x,y)+g(x,y)] = 0

Teorema:

Sea P[f(x,y)] = d_{x}[ f(0,0) ]+d_{y}[f(0,0)] ==> ...

... f(x,y) = (-k)·( e^{x}+e^{y} ) es vector dual de g(x,y) = (3/2)·k·(x+y)

Teorema:

Sea Q[f(x,y)] = d_{x}[ f(0,0) ]+d_{y}[ f(0,0) ] ==> ...

... f(x,y) = (-k)·( sin(x)+sin(y) ) es vector dual de g(x,y) = (3/2)·k·(x+y)

Teorema:

Sea A[f(x,y)] = int[x = 0]-[pi·i][ f(x,0) ]d[x]+int[y = 0]-[pi·i][ f(0,y) ]d[y] ==> ...

... f(x,y) = (-k)·( e^{x}+e^{y} ) es vector dual de g(x,y) = (1/pi)^{2}·7k·(x+y)

Teorema:

Sea B[f(x,y)] = int[x = 0]-[pi][ f(x,0) ]d[x]+int[y = 0]-[pi][ f(0,y) ]d[y] ==> ...

... f(x,y) = k·( sin(x)+sin(y) ) es vector dual de g(x,y) = (-1)·(1/pi)^{2}·7k·(x+y)

Análisis funcional:

Operador Sturm-Lioville:

Arte:

[EP(x)][EQ(x)][ d_{x}[ P(x)·d_{x}[f(x)] ]+Q(x)·f(x) = 0 <==> ...

... d_{x}[P(x)]·d_{x}[f(x)]+Q(x)·f(x) = e^{i·int[ ( 1/P(x) ) ]d[x]} ]

[EP(x)][EQ(x)][ d_{x}[ P(x)·d_{x}[f(x)] ]+Q(x)·f(x) = 0 <==> ...

... d_{x}[P(x)]·d_{x}[f(x)]+Q(x)·f(x) = e^{(-i)·int[ ( 1/P(x) ) ]d[x]} ]

Exposición:

P(x) = 1 & Q(x) = 1

d_{xx}^{2}[f(x)]+f(x) = 0

f(x) = e^{ix} || f(x) = e^{(-i)·x}

Wronsk( d_{x}[f(x)] ) = | < d_{x}[P(x)],i·Q(x) >,< i·f(x),d_{x}[f(x)] > |

P(x)·d_{xx}^{2}[f(x)] = (-1)·Wronsk( d_{x}[f(x)] )

h( d_{d_{x}[f(x)]}[ Wronsk( d_{x}[f(x)] ) ] ) = i

ln( Wronsk( d_{x}[f(x)] ) ) = i·int[ ( 1/P(x) ) ]d[x]

Wronsk( d_{x}[f(x)] ) = e^{i·int[ ( 1/P(x) ) ]d[x]}

d_{x}[P(x)]·d_{x}[f(x)]+Q(x)·f(x) = e^{i·int[ ( 1/P(x) ) ]d[x]}

h( d_{d_{x}[f(x)]}[ Wronsk( d_{x}[f(x)] ) ] ) = (-i)

ln( Wronsk( d_{x}[f(x)] ) ) = (-i)·int[ ( 1/P(x) ) ]d[x]

Wronsk( d_{x}[f(x)] ) = e^{(-i)·int[ ( 1/P(x) ) ]d[x]}

d_{x}[P(x)]·d_{x}[f(x)]+Q(x)·f(x) = e^{(-i)·int[ ( 1/P(x) ) ]d[x]}