integrals

∫ [ ( ln(x) )^{n} ] d[x] = (1/(n+1))·( ln(x) )^{n+1} [o(x)o] (1/2)·x^{2}


∫ [ ( ln( f(x) ) )^{n} ] d[x] = ...
... (1/(n+1))·( ln( f(x) ) )^{n+1} [o(x)o] ∫ [ f(x) ] d[x] [o(x)o] ( f(x) )^{[o(x)o](-1)}


∫ [ ( ln( ax^{2}+bx ) )^{n} ] d[x] = ...
... (1/(n+1))·( ln( ax^{2}+bx ) )^{n+1} [o(x)o] ( a·(1/3)·x^{3}+b·(1/2)·x^{2} ) [o(x)o] (1/2a)·ln( 2ax+b )

psíquica depresión

depresión para-noide: de nodo o centro.
desactivación del centro del campo de proyección de objetivos
no salir de casa
no ducharse
no cocinar

psíquica esquizofrenia

esquizofrenia para-noide: de nodo, centro o chakra.
activación de centros de ondas.
alucinación de imágenes.
alucinación de sonidos.

series numériques criteri

teorema:
Si [∀k][ k€N ==> [∃s][ s > 0 & a_{k} [< s^{k} < 1 & (a_{k}/a_{k+1}) [< a_{k+(-1)} ] ] ==> ...
... ∑ a_{k} [< ( 1/( 1+(-1)·s^{2} ) ) [< oo


demostració:
(a_{k}/a_{k+1}) [< a_{k+(-1)}
a_{k} [< a_{k+1}a_{k+(-1)} [< s^{k+1}s^{k+(-1)} = s^{2k}


∑ a_{k} [< ∑ s^{2k} [< ( 1/( 1+(-1)·s^{2} ) ) [< oo


teorema:
Si [∀k][ k€N ==> [∃s][ s > 0 & a_{k} [< s^{k} < 1 & (a_{k}/a_{k+p}) [< a_{k+(-p)} ] ] ==> ...
... ∑ a_{k} [< ( 1/( 1+(-1)·s^{2} ) ) [< oo

series numériques criteri del quocient

teorema:
Si [∀k][ k€N ==> [∃s][ s > 0 & a_{1} [< 1 & ( a_{k+1}/a_{k} ) [< s [< ( 1+(-1)(1/oo) ) ] ] ==> ...
... ∑ a_{k} [< oo


demostració:
( a_{k+1}/a_{k} ) [< s
a_{k+1} [< s·a_{k}
a_{k+1} [< s^{k+1}
∑ a_{k} [< ∑ s^{k} [< ( 1/(1+(-s)) ) [< oo


teorema:
( k/(k+1) ) = ( 1+( (-1)/(k+1) ) ) [< ( 1+(-1)(1/oo) ) = s


∑ ( 1/k ) [< oo

series numériques criteri de la arrel enéssima

teorema:
Si [∀k][ k€N ==> [∃s][ s > 0 & ( a_{k} )^{(1/k)} [< s [< ( 1+(-1)(1/oo) ) ] ] ==> ∑ a_{k} [< oo


demostració:
∑ a_{k} [< ∑ s^{k} [< ( 1/(1+(-s)) ) [< oo

analisis funcional formes integrals

∫ [0-->1]-[ ( (-1)·( k/a )·x+( (2k)/b ) ) [o] (ax+b) ] d[x] = k


∫ [0-->1]-[ ( (-1)·( k/a )·e^{x}+( k/b )·e^{2} ) [o] (a·e^{x}+b) ] d[x] = k


∫ [0-->1]-[ ( ( (12k)/(7a) )·( 1/(x+1)^{2} )+( k/b )·( 1/(x+1) ) ) [o] ...
... ( a·( 1/(x+1)^{2} )+b·( 1/(x+1) ) ) ] d[x] = k

operadors ortogonals autoadjunts

F(P(x)) [o] G(P(x)) = 0
(-1)·F(P(x)) [o] G(P(x)) = 0
F(P(x)) [o] (-1)·G(P(x)) = 0
(-1)·F(P(x)) [o] (-1)·G(P(x)) = 0


F(a·e^{2x}+b·e^{x}) = 2a·e^{2x}+b·e^{x}
G(a·e^{2x}+b·e^{x}) = (-1)( 1/(2a) )·e^{2x}+(1/b)·e^{x}

operadors ortogonals autoadjunts

F(P(x)) [o] G(P(x)) = 0
(-1)·F(P(x)) [o] G(P(x)) = 0
F(P(x)) [o] (-1)·G(P(x)) = 0
(-1)·F(P(x)) [o] (-1)·G(P(x)) = 0


F(ax^{2}+bx) = 2ax+b
G(ax^{2}+bx) = (-1)( 1/(2a) )·x+(1/b)


F(ax^{3}+bx^{2}+cx) = 3ax^{2}+2bx+c
G(ax^{3}+bx^{2}+cx) = (-1)( 1/(3a) )·x^{2}+(-1)( 1/(2b) )·x+(2/c)


F(ax^{2}+bx+c) = (a/3)x^{3}+(b/2)x^{2}+cx
G(ax^{2}+bx+c) = (-1)( 3/a )·x^{3}+(-1)( 2/b )·x^{2}+(2/c)·x