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# Formulario de calculo integral y diferencial

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### Formulario de calculo integral y diferencial

1. 1. http://www.geocities.com/calculusjrm/ Fórmulas de Cálculo Diferencial e Integral (Página 1 de 3) Fórmulas de Cálculo Diferencial e Integral VER.6.8 Jesús Rubí Miranda (jesusrubim@yahoo.com) http://www.geocities.com/calculusjrm/ VALOR ABSOLUTO ( a + b ) ⋅ ( a 2 − ab + b 2 ) = a 3 + b3 ( a + b ) ⋅ ( a3 − a 2 b + ab 2 − b3 ) = a 4 − b 4 ( a + b ) ⋅ ( a 4 − a 3b + a 2 b 2 − ab3 + b 4 ) = a 5 + b5 ( a + b ) ⋅ ( a5 − a 4 b + a 3b 2 − a 2 b3 + ab 4 − b5 ) = a 6 − b 6 ⎛ n ⎞ k +1 ( a + b ) ⋅ ⎜ ∑ ( −1) a n− k b k −1 ⎟ = a n + b n ∀ n ∈ ⎝ k =1 ⎠ ⎛ ⎞ a n − k b k −1 ⎟ = a n − b n ∀ n ∈ ⎝ k =1 ⎠ SUMAS Y PRODUCTOS a = −a a1 + a2 + a ≤ a y −a ≤ a a ≥0 y a =0 ⇔ a=0 ab = a b ó n a+b ≤ a + b ó k n n ≤ ∑ ak k k =1 (a ⋅b) =a n k =1 k =1 n k =1 ap = a p−q aq p k =1 n ∑(a k =1 = a ⋅b p p ap ⎛a⎞ ⎜ ⎟ = p b ⎝b⎠ a p/q = a p q LOGARITMOS log a N = x ⇒ a x = N log a MN = log a M + log a N M = log a M − log a N N log a N r = r log a N log a log b N ln N = log a N = log b a ln a 1+ 3 + 5 + log10 N = log N y log e N = ln N π π , 2 2 tg 2 θ + 1 = sec 2 θ sin ( −θ ) = − sin θ sin θ + cos 2 θ = 1 2 1 + ctg 2 θ = csc 2 θ ( a + b) ⋅ ( a − b) = a − b 2 ( a + b ) ⋅ ( a + b ) = ( a + b ) = a 2 + 2ab + b 2 2 ( a − b ) ⋅ ( a − b ) = ( a − b ) = a 2 − 2ab + b 2 ( x + b ) ⋅ ( x + d ) = x 2 + ( b + d ) x + bd ( ax + b ) ⋅ ( cx + d ) = acx 2 + ( ad + bc ) x + bd ( a + b ) ⋅ ( c + d ) = ac + ad + bc + bd 3 ( a + b ) = a3 + 3a 2b + 3ab 2 + b3 3 ( a − b ) = a 3 − 3a 2b + 3ab 2 − b3 2 ( a + b + c ) = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc 2 ( a − b ) ⋅ ( a + ab + b ) = a − b ( a − b ) ⋅ ( a 3 + a 2 b + ab 2 + b3 ) = a 4 − b 4 ( a − b ) ⋅ ( a 4 + a 3b + a 2 b 2 + ab3 + b 4 ) = a 5 − b5 2 n 2 ⎞ 3 3 ( a − b ) ⋅ ⎜ ∑ a n − k b k −1 ⎟ = a n − b n ⎝ k =1 ⎠ ∀n ∈ tg (θ + π ) = tg θ sen x cos x tg x -1.5 -2 0 2 4 6 sin (θ + nπ ) = ( −1) sin θ n 8 Gráfica 2. Las funciones trigonométricas csc x , sec x , ctg x : 2.5 1 0.5 2 0 -0.5 -1 k =1 ( x1 + x2 + + xk ) = ∑ n -1.5 -2 0 2 4 6 8 xknk 4 n ⎛ 2n + 1 ⎞ sin ⎜ π ⎟ = ( −1) ⎝ 2 ⎠ ⎛ 2n + 1 ⎞ cos ⎜ π⎟=0 ⎝ 2 ⎠ ⎛ 2n + 1 ⎞ tg ⎜ π⎟=∞ ⎝ 2 ⎠ cosh : tgh : ctgh : → → [1, ∞ → −1,1 − {0} → −∞ , −1 ∪ 1, ∞ sech : → 0 ,1] csch : − {0} → -1 arc sen x arc cos x arc tg x -2 -1 0 1 2 3 cos 2θ = cos 2 θ − sin 2 θ 2 tg θ tg 2θ = 1 − tg 2 θ 1 sin 2 θ = (1 − cos 2θ ) 2 1 cos 2 θ = (1 + cos 2θ ) 2 1 − cos 2θ tg 2 θ = 1 + cos 2θ − {0} Gráfica 5. Las funciones hiperbólicas sinh x , cosh x , tgh x : 5 4 π⎞ ⎛ sin θ = cos ⎜ θ − ⎟ 2⎠ ⎝ π⎞ ⎛ cos θ = sin ⎜ θ + ⎟ 2⎠ ⎝ tg α ± tg β tg (α ± β ) = 1 ∓ tg α tg β sin 2θ = 2sin θ cos θ 0 CO sinh : n 3 2 1 0 -1 -2 cos (α ± β ) = cos α cos β ∓ sin α sin β 1 -2 -3 tg (θ + nπ ) = tg θ sin ( nπ ) = 0 sin (α ± β ) = sin α cos β ± cos α sin β 2 π radianes=180 CA -4 3 e = 2.71828182846… TRIGONOMETRÍA CO 1 sen θ = cscθ = HIP sen θ CA 1 cosθ = secθ = HIP cosθ sen θ CO 1 tgθ = ctgθ = = cosθ CA tgθ θ -6 Gráfica 3. Las funciones trigonométricas inversas arcsin x , arccos x , arctg x : CONSTANTES π = 3.14159265359… HIP csc x sec x ctg x -2 n! n x1n1 ⋅ x2 2 n1 ! n2 ! nk ! n tg ( nπ ) = 0 1.5 -2.5 -8 cos (θ + nπ ) = ( −1) cos θ cos ( nπ ) = ( −1) 2 ex − e− x 2 e x + e− x cosh x = 2 sinh x e x − e − x = tgh x = cosh x e x + e− x e x + e− x 1 = ctgh x = tgh x e x − e − x 1 2 = sech x = cosh x e x + e − x 1 2 = csch x = sinh x e x − e − x sinh x = cos (θ + π ) = − cos θ -1 ⎛n⎞ n! , k≤n ⎜ ⎟= ⎝ k ⎠ ( n − k )!k ! n ⎛n⎞ n ( x + y ) = ∑ ⎜ ⎟ xn−k y k k =0 ⎝ k ⎠ tg ( −θ ) = − tg θ tg (θ + 2π ) = tg θ -0.5 sin α ⋅ cos β = tg α + tg β ctg α + ctg β FUNCIONES HIPERBÓLICAS sin (θ + π ) = − sin θ -4 5 cos ( −θ ) = cos θ cos (θ + 2π ) = cos θ 0 n ALGUNOS PRODUCTOS a ⋅ ( c + d ) = ac + ad 2 + ( 2n − 1) = n tg α ⋅ tg β = y ∈ 0, π sin (θ + 2π ) = sin θ -6 sin (α ± β ) cos α ⋅ cos β 1 ⎡sin (α − β ) + sin (α + β ) ⎤ ⎦ 2⎣ 1 sin α ⋅ sin β = ⎡cos (α − β ) − cos (α + β ) ⎤ ⎦ 2⎣ 1 cos α ⋅ cos β = ⎡cos (α − β ) + cos (α + β ) ⎤ ⎦ 2⎣ 0 IDENTIDADES TRIGONOMÉTRICAS 0.5 -2 -8 tg α ± tg β = arc ctg x arc sec x arc csc x -2 -5 1 n! = ∏ k ⎛ y∈ − Jesús Rubí M. 1 1 (α + β ) ⋅ cos (α − β ) 2 2 1 1 sin α − sin β = 2 sin (α − β ) ⋅ cos (α + β ) 2 2 1 1 cos α + cos β = 2 cos (α + β ) ⋅ cos (α − β ) 2 2 1 1 cos α − cos β = −2 sin (α + β ) ⋅ sin (α − β ) 2 2 sin α + sin β = 2sin 0 1.5 p 2 -1 n pq 3 1 2 n ∑ ⎡ a + ( k − 1) d ⎤ = 2 ⎡ 2a + ( n − 1) d ⎤ ⎣ ⎦ ⎣ ⎦ k =1 n = (a + l ) 2 n 1 − r n a − rl k −1 ∑ ar = a 1 − r = 1 − r k =1 n 1 ∑ k = 2 ( n2 + n ) k =1 n 1 ∑ k 2 = 6 ( 2n3 + 3n2 + n ) k =1 n 1 ∑ k 3 = 4 ( n 4 + 2n3 + n 2 ) k =1 n 1 ∑ k 4 = 30 ( 6n5 + 15n4 + 10n3 − n ) k =1 4 y ∈ ⎢− , ⎥ ⎣ 2 2⎦ y = ∠ cos x y ∈ [ 0, π ] Gráfica 1. Las funciones trigonométricas: sin x , cos x , tg x : − ak −1 ) = an − a0 k y = ∠ sin x 1 y = ∠ sec x = ∠ cos y ∈ [ 0, π ] x 1 ⎡ π π⎤ y = ∠ csc x = ∠ sen y ∈ ⎢− , ⎥ x ⎣ 2 2⎦ ∑ ( ak + bk ) = ∑ ak + ∑ bk EXPONENTES (a ) k =1 Gráfica 4. Las funciones trigonométricas inversas arcctg x , arcsec x , arccsc x : ⎡ π π⎤ 1 y = ∠ ctg x = ∠ tg x = c ∑ ak n k =1 a p ⋅ a q = a p+q p q + an = ∑ ak n k k =1 tg ctg cos sec csc sin 0 0 ∞ ∞ 1 1 12 3 2 3 2 3 2 1 3 1 2 1 2 2 2 1 1 3 1 3 2 2 3 3 2 12 0 0 ∞ ∞ 1 1 y = ∠ tg x n k =1 ∑ ca k =1 ∑a par ∑ c = nc = ∏ ak n k +1 n n ∏a k =1 n ( a + b ) ⋅ ⎜ ∑ ( −1) ⎧a si a ≥ 0 a =⎨ ⎩− a si a < 0 impar θ 0 30 45 60 90 senh x cosh x tgh x -3 -4 -5 0 5 FUNCIONES HIPERBÓLICAS INV ( ( ) ) sinh −1 x = ln x + x 2 + 1 , ∀x ∈ cosh −1 x = ln x ± x 2 − 1 , x ≥ 1 tgh −1 x = 1 ⎛1+ x ⎞ ln ⎜ ⎟, 2 ⎝1− x ⎠ 1 ⎛ x +1⎞ ctgh −1 x = ln ⎜ ⎟, 2 ⎝ x −1⎠ x <1 x >1 ⎛ 1 ± 1 − x2 ⎞ ⎟, 0 < x ≤ 1 sech −1 x = ln ⎜ ⎜ ⎟ x ⎝ ⎠ ⎛1 x2 + 1 ⎞ ⎟, x ≠ 0 csch −1 x = ln ⎜ + ⎜x x ⎟ ⎝ ⎠
2. 2. http://www.geocities.com/calculusjrm/ Fórmulas de Cálculo Diferencial e Integral (Página 2 de 3) IDENTIDADES DE FUNCS HIP cosh 2 x − sinh 2 x = 1 1 − tgh 2 x = sech 2 x ctgh 2 x − 1 = csch 2 x sinh ( − x ) = − sinh x cosh ( − x ) = cosh x tgh ( − x ) = − tgh x sinh ( x ± y ) = sinh x cosh y ± cosh x sinh y cosh ( x ± y ) = cosh x cosh y ± sinh x sinh y tgh x ± tgh y 1 ± tgh x tgh y sinh 2 x = 2sinh x cosh x tgh ( x ± y ) = cosh 2 x = cosh 2 x + sinh 2 x 2 tgh x tgh 2 x = 1 + tgh 2 x 1 sinh 2 x = ( cosh 2 x − 1) 2 1 cosh 2 x = ( cosh 2 x + 1) 2 cosh 2 x − 1 tgh 2 x = cosh 2 x + 1 sinh 2 x tgh x = cosh 2 x + 1 e x = cosh x + sinh x e − x = cosh x − sinh x OTRAS ax 2 + bx + c = 0 −b ± b 2 − 4ac 2a b 2 − 4ac = discriminante ⇒ x= exp (α ± i β ) = e α ( cos β ± i sin β ) si α , β ∈ LÍMITES 1 lim (1 + x ) x = e = 2.71828... x →0 x ⎛ 1⎞ lim ⎜1 + ⎟ = e x →∞ ⎝ x⎠ sen x lim =1 x →0 x 1 − cos x lim =0 x →0 x ex −1 lim =1 x →0 x x −1 lim =1 x →1 ln x DERIVADAS f ( x + ∆x ) − f ( x ) df ∆y Dx f ( x ) = = lim = lim ∆x → 0 ∆x dx ∆x →0 ∆x d (c) = 0 dx d ( cx ) = c dx d ( cx n ) = ncx n−1 dx d du dv dw (u ± v ± w ± ) = ± ± ± dx dx dx dx d du ( cu ) = c dx dx d dv du ( uv ) = u + v dx dx dx d dw dv du ( uvw ) = uv + uw + vw dx dx dx dx d ⎛ u ⎞ v ( du dx ) − u ( dv dx ) ⎜ ⎟= dx ⎝ v ⎠ v2 d n n −1 du ( u ) = nu dx dx dF dF du = ⋅ (Regla de la Cadena) dx du dx du 1 = dx dx du dF dF du = dx dx du ⎧ dy dy dt f 2′ ( t ) ⎪ x = f1 ( t ) = = donde ⎨ dx dx dt f1′( t ) ⎪ y = f2 (t ) ⎩ DERIVADA DE FUNCS LOG & EXP d du dx 1 du = ⋅ ( ln u ) = dx u u dx d log e du ⋅ ( log u ) = dx u dx log e du d ( log a u ) = a ⋅ a > 0, a ≠ 1 dx u dx d u ( e ) = eu ⋅ du dx dx d u ( a ) = au ln a ⋅ du dx dx d v ( u ) = vu v−1 du + ln u ⋅ u v ⋅ dv dx dx dx DERIVADA DE FUNCIONES TRIGO d du ( sin u ) = cos u dx dx d du ( cos u ) = − sin u dx dx d du ( tg u ) = sec2 u dx dx d du ( ctg u ) = − csc2 u dx dx d du ( sec u ) = sec u tg u dx dx d du ( csc u ) = − csc u ctg u dx dx d du ( vers u ) = sen u dx dx DERIV DE FUNCS TRIGO INVER 1 d du ⋅ ( ∠ sin u ) = dx 1 − u 2 dx 1 d du ⋅ ( ∠ cos u ) = − dx 1 − u 2 dx 1 d du ⋅ ( ∠ tg u ) = dx 1 + u 2 dx 1 d du ⋅ ( ∠ ctg u ) = − dx 1 + u 2 dx 1 d du ⎧+ si u > 1 ⋅ ⎨ ( ∠ sec u ) = ± dx u u 2 − 1 dx ⎩− si u < −1 1 d du ⎧− si u > 1 ⋅ ⎨ ( ∠ csc u ) = ∓ dx u u 2 − 1 dx ⎩+ si u < −1 1 d du ⋅ ( ∠ vers u ) = dx 2u − u 2 dx DERIVADA DE FUNCS HIPERBÓLICAS d du sinh u = cosh u dx dx d du cosh u = sinh u dx dx d du tgh u = sech 2 u dx dx d du ctgh u = − csch 2 u dx dx d du sech u = − sech u tgh u dx dx d du csch u = − csch u ctgh u dx dx DERIVADA DE FUNCS HIP INV d du 1 senh −1 u = ⋅ dx 1 + u 2 dx d du ±1 cosh −1 u = ⋅ , u >1 dx u 2 − 1 dx d 1 du ⋅ , u <1 tgh −1 u = dx 1 − u 2 dx d 1 du −1 ⋅ , u >1 ctgh u = dx 1 − u 2 dx d du ⎧− si ∓1 ⎪ sech −1 u = ⋅ ⎨ dx u 1 − u 2 dx ⎪ + si ⎩ ⎧+ si cosh -1u > 0 ⎪ ⎨ -1 ⎪− si cosh u < 0 ⎩ sech −1 u > 0, u ∈ 0,1 sech −1 u < 0, u ∈ 0,1 ∫ { f ( x ) ± g ( x )} dx = ∫ f ( x ) dx ± ∫ g ( x ) dx ∫ cf ( x ) dx = c ⋅ ∫ f ( x ) dx c ∈ ∫ f ( x ) dx = ∫ f ( x ) dx + ∫ f ( x ) dx ∫ f ( x ) dx = − ∫ f ( x ) dx ∫ f ( x ) dx = 0 m ⋅ ( b − a ) ≤ ∫ f ( x ) dx ≤ M ⋅ ( b − a ) b a b a b a b a b c b a a c b a a b a a b a ⇔ m ≤ f ( x ) ≤ M ∀x ∈ [ a, b ] , m, M ∈ ∫ f ( x ) dx ≤ ∫ g ( x ) dx b b a a ⇔ f ( x ) ≤ g ( x ) ∀x ∈ [ a , b ] ∫ f ( x ) dx ≤ ∫ f ( x ) dx si a < b b b a a INTEGRALES ∫ adx =ax ∫ af ( x ) dx = a ∫ f ( x ) dx ∫ ( u ± v ± w ± ) dx = ∫ udx ± ∫ vdx ± ∫ wdx ± ∫ udv = uv − ∫ vdu ( Integración por partes ) u n+1 ∫ u du = n + 1 n du ∫u = ln u u u au ⎧a > 0 u ∫ a du = ln a ⎨a ≠ 1 ⎩ au ⎛ −1 1 ⎞ ∫ ua du = ln a ⋅ ⎜ u − ln a ⎟ ⎝ ⎠ u ∫ ue du = e ( u − 1) ∫ ln udu =u ln u − u = u ( ln u − 1) u u 1 u ( u ln u − u ) = ( ln u − 1) ln a ln a u2 ∫ u log a udu = 4 ⋅ ( 2log a u − 1) u2 ∫ u ln udu = 4 ( 2ln u − 1) INTEGRALES DE FUNCS TRIGO ∫ log a udu = ∫ sin udu = − cos u ∫ cos udu = sin u ∫ sec udu = tg u ∫ csc udu = − ctg u ∫ sec u tg udu = sec u ∫ csc u ctg udu = − csc u ∫ tg udu = − ln cos u = ln sec u ∫ ctg udu = ln sin u ∫ sec udu = ln sec u + tg u ∫ csc udu = ln csc u − ctg u 1 = ln tgh u 2 INTEGRALES DE FRAC du 1 u ∫ u 2 + a 2 = a ∠ tg a u 1 = − ∠ ctg a a du 1 u−a 2 2 ∫ u 2 − a 2 = 2a ln u + a ( u > a ) du 1 a+u 2 2 ∫ a 2 − u 2 = 2a ln a − u ( u < a ) INTEGRALES CON ∫ du = ∠ sin a2 − u2 ∫ ∫ ctg 2 2 ( du = ln u + u 2 ± a 2 u 2 ± a2 ( ∫e udu = − ( ctg u + u ) ∫ u sin udu = sin u − u cos u ∫e ∫ u cos udu = cos u + u sin u au au sin bu du = cos bu du = INTEGRALES DE FUNCS TRIGO INV 2 2 a 2 + b2 e au ( a cos bu + b sin bu ) a2 + b2 1 1 ∫ sec u du = 2 sec u tg u + 2 ln sec u + tg u ALGUNAS SERIES + + 2 2 −1 = u∠ sec u − ∠ cosh u ∫ ∠ csc udu = u∠ csc u + ln ( u + u2 − 1 ) ) = u∠ csc u + ∠ cosh u INTEGRALES DE FUNCS HIP 2 e au ( a sin bu − b cos bu ) f '' ( x0 )( x − x0 ) f ( x ) = f ( x0 ) + f ' ( x0 )( x − x0 ) + 2 2 ) 3 ∫ ∠ sin udu = u∠ sin u + 1 − u ∫ ∠ cos udu = u∠ cos u − 1 − u ∫ ∠ tg udu = u∠ tg u − ln 1 + u ∫ ∠ ctg udu = u∠ ctg u + ln 1 + u ∫ ∠ sec udu = u∠ sec u − ln ( u + u ∫ sinh udu = cosh u ∫ cosh udu = sinh u ∫ sech udu = tgh u ∫ csch udu = − ctgh u ∫ sech u tgh udu = − sech u ∫ csch u ctgh udu = − csch u ) 1 u ∫ u a 2 ± u 2 = a ln a + a 2 ± u 2 1 du a ∫ u u 2 − a 2 = a ∠ cos u 1 u = ∠ sec a a u 2 a2 u 2 2 2 ∫ a − u du = 2 a − u + 2 ∠ sen a 2 u 2 a 2 2 2 2 2 ∫ u ± a du = 2 u ± a ± 2 ln u + u ± a MÁS INTEGRALES udu = n ≠ −1 u a du u 1 − sin 2u 2 4 u 1 2 ∫ cos udu = 2 + 4 sin 2u 2 ∫ tg udu = tg u − u ∫ sin u a = −∠ cos 2 INTEGRALES DEFINIDAS, PROPIEDADES Nota. Para todas las fórmulas de integración deberá agregarse una constante arbitraria c (constante de integración). b ∫ e du = e Jesús Rubí M. ∫ tgh udu = ln cosh u ∫ ctgh udu = ln sinh u ∫ sech udu = ∠ tg ( sinh u ) ∫ csch udu = − ctgh ( cosh u ) 2 d du 1 csch −1 u = − ⋅ , u≠0 dx u 1 + u 2 dx a INTEGRALES DE FUNCS LOG & EXP f ( n ) ( x0 )( x − x0 ) n! f ( x ) = f (0) + f ' ( 0) x + + + f ( n) ( 0 ) x n 2! n : Taylor f '' ( 0 ) x 2 2! : Maclaurin n! x 2 x3 xn + + + + n! 2! 3! x 3 x5 x 7 x 2 n −1 n −1 − + + ( −1) sin x = x − + 3! 5! 7! ( 2n − 1)! ex = 1 + x + cos x = 1 − x2 x4 x6 + + − 2! 4! 6! + ( −1) n −1 x 2 n− 2 ( 2n − 2 ) ! n x2 x3 x 4 n −1 x + − + + ( −1) n 2 3 4 2 n −1 x3 x 5 x 7 n −1 x ∠ tg x = x − + − + + ( −1) 3 5 7 2n − 1 ln (1 + x ) = x − 2
3. 3. Fórmulas de Cálculo Diferencial e Integral (Página 3 de 3) ALFABETO GRIEGO Mayúscula Minúscula Nombre 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Α Β Γ ∆ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω α β γ δ ε ζ η θ ϑ ι κ λ µ ν ξ ο π ϖ ρ ς τ υ φ ϕ χ ψ ω σ Alfa Beta Gamma Delta Epsilon Zeta Eta Teta Iota Kappa Lambda Mu Nu Xi Omicron Pi Rho Sigma Tau Ipsilon Phi Ji Psi Omega Equivalente Romano A B G D E Z H Q I K L M N X O P R S T U F C Y W NOTACIÓN sin cos tg Seno. Coseno. Tangente. sec csc ctg Secante. Cosecante. Cotangente. vers Verso seno. arcsin θ = sin θ Arco seno de un ángulo θ . u = f ( x) sinh Seno hiperbólico. cosh Coseno hiperbólico. tgh Tangente hiperbólica. ctgh Cotangente hiperbólica. sech Secante hiperbólica. csch Cosecante hiperbólica. u, v, w Funciones de x , u = u ( x ) , v = v ( x ) . Conjunto de los números reales. = {…, −2, −1,0,1, 2,…} Conjunto de enteros. Conjunto de números racionales. c Conjunto de números irracionales. = {1, 2,3,…} Conjunto de números naturales. Conjunto de números complejos. http://www.geocities.com/calculusjrm/ Jesús Rubí M.