Hyperbolic Functions (A) 08/3/04
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Parabola Parabola From Ancient Greek παραβολή (parabolē), from παραβάλλω (paraballō, I set side by side ), from παρά (para, beside ) + βάλλω (ballō, I throw ). The conic section formed by the intersection of a cone with a plane parallel to a tangent plane to the cone; the locus of points equidistant from a fixed point (the focus) and line (the directrix). http://en.wikipedia.org/ Hyperbolic Function (A) 3 08/3/04
Hyperbola Hyperbola From ὐπερβάλλω "I go beyond, exceed", from ὐπέρ "above" + βάλλω "I throw" In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve, lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. If the plane intersects both halves of the double cone but does not pass through the apex of the cones then the conic is a hyperbola. A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone. http://en.wikipedia.org/ Hyperbolic Function (A) 4 08/3/04
Trigonometric & Hyperbolic Functions cos α + sin α = cosh α sinh α = http://en.wikipedia.org/ Hyperbolic Function (A) 5 08/3/04
cosh(x) +α α cosh α = (e + e ) cos θ = + jθ (e + e j θ ) http://en.wikipedia.org/ Hyperbolic Function (A) 6 08/3/04
sinh(x) sinh α = sin θ = +α α (e e ) + jθ (e e j θ ) j http://en.wikipedia.org/ Hyperbolic Function (A) 7 08/3/04
Definitions of Hyperbolic Functions cosh = e e sinh = e e e e tanh = e e http://en.wikipedia.org/ Hyperbolic Function (A) 8 08/3/04
Definitions of Hyperbolic Functions cosh = e e x y = sinh = e e cosh sinh = e e tanh = e e cosh, sinh x y α (e + e α ) (eα e α ) = 4 4 http://en.wikipedia.org/ Hyperbolic Function (A) 9 08/3/04
Hyperbolic vs. Trigonometric Functions Trigonometric Function Hyperbolic Function x ix e +i x = cos x + i sin x e + x = cosh x + sinh x e i x = cos x i sin x e x = cosh x sinh x cos x = +ix (e + e ix ) cosh x = +x x (e + e ) sin x = +ix (e e ix ) i sinh x = +x (e e x ) + ix ix (e e ) tan x = i (e+ix + e ix ) Hyperbolic Function (A) (e+ x e x ) tanh x = (e+ x + e x ) 0 08/3/04
Trigonometric functions with imaginary arguments ix x cos x = +ix (e + e ix ) cosh x = +x x (e + e ) sin x = +ix (e e ix ) i sinh x = +x x (e e ) + ix ix (e e ) tan x = i (e+ix + e ix ) (e+ x e x ) tanh x = (e+ x + e x ) cos i x = cosh x cos i x = x +x (e + e ) cosh x = +x x (e + e ) sin i x = i sinh x sin i x = x (e e + x ) i sinh x = +x x (e e ) tan i x = i tanh x x +x (e e ) tan i x = i (e x + e + x ) Hyperbolic Function (A) (e+ x e x ) tanh x = (e+ x + e x ) 08/3/04
Hyperbolic functions with imaginary arguments cos x = +ix (e + e ix ) cosh x = +x x (e + e ) sin x = +ix (e e ix ) i sinh x = +x x (e e ) + ix ix (e e ) tan x = i (e+ix + e ix ) x ix (e+ x e x ) tanh x = (e+ x + e x ) cos x = (e +ix + e ix ) cosh i x = +i x i x (e + e ) cosh i x = cos x +ix ix sin x = (e e ) i sinh i x = +i x (e e i x ) sinh i x = i sin x + ix ix (e e ) tan x = i (e+ix + e ix ) (e+i x e i x ) tanh i x = (e +i x + e i x ) Hyperbolic Function (A) tanh i x = i tan x 08/3/04
With imaginary arguments cos x = +ix (e + e ix ) cosh x = +x x (e + e ) sin x = +ix (e e ix ) i sinh x = +x x (e e ) + ix ix (e e ) tan x = i (e+ix + e ix ) (e+ x e x ) tanh x = (e+ x + e x ) cos i x = cosh x cosh i x = cos x sin i x = i sinh x sinh i x = i sin x tan i x = i tanh x tanh i x = i tan x Hyperbolic Function (A) x 3 ix 08/3/04
Euler Formula Euler Formula Euler Formula e +i x = cos x + i sin x e +i x = cosh i x + sinh i x e i x = cos x i sin x e i x = cosh i x sinh i x cos i x = cosh x cosh i x = cos x sin i x = i sinh x sinh i x = i sin x tan i x = i tanh x tanh i x = i tan x Hyperbolic Function (A) 4 08/3/04
Modulus of sin(z) () sin (z) = sin( x +i y ) = sin( x)cos (i y) + cos (x )sin(i y ) = sin( x)cosh ( y ) + i cos( x)sinh ( y ) sin ( z) = sin (z) sin( z) = i = 4 (e y +i x e+ y i x )(e y i x e + y +i x ) = 4 (e y e+ i x e i x + e+ y ) = 4 (e+ y + e y e + i x + e i x ) (e +i (x +i y ) e i( x +i y ) i ) (e i (x i y) e + i( x i y ) ) = + 4 (e+ y + e y ) 4 (e+ i x + e i x ) = [ (e + y e y ) ] + [ i (e+i x e i x ) ] = sin (x ) + sinh ( y ) Hyperbolic Function (A) 5 08/3/04
Modulus of sin(z) () sin (z) = sin( x +i y ) = sin( x)cos (i y) + cos (x )sin(i y ) = sin( x)cosh ( y ) + i cos( x)sinh ( y ) sin ( z) = sin( x) cosh( y ) + i cos( x )sinh( y) = sin (x )cosh ( y) + cos ( x)sinh ( y ) cosh sinh = = sin (x )( + sinh ( y )) + ( sin ( x))sinh ( y) cos α + sin α = = sin (x ) + sin (x)sinh ( y) + sinh ( y ) sin ( x)sinh ( y ) = sin (x ) + sinh ( y ) Hyperbolic Function (A) 6 08/3/04
Graphs of sin(z) sin (x )cosh ( y ) sin (x) cosh ( y) sin ( x) + sinh ( y ) i cos(x )sinh ( y ) Re{sin(z)} sin(z) Im{sin(z)} cos (x ) sinh ( y) x y sin (z) = sin( x +i y ) = sin( x)cosh ( y ) + i cos( x)sinh ( y ) sin ( x ) sin ( z) = sin ( x ) + sinh ( y ) sinh ( y) http://en.wikipedia.org/ Hyperbolic Function (A) 7 08/3/04
Domain Coloring Argument Domain z Coloring f ( z) Argument of f(z) http://en.wikipedia.org/ Hyperbolic Function (A) 8 08/3/04
Domain Coloring Modulus Domain z Coloring f ( z) Modulus of f(z) http://en.wikipedia.org/ Hyperbolic Function (A) 9 08/3/04
Domain Coloring of sin(z) Domain coloring of sin(z) over (-π,π) on x and y axes. Brightness indicates absolute magnitude, saturation represents imaginary and real magnitude. sin ( x) + sinh ( y ) http://en.wikipedia.org/ Hyperbolic Function (A) 0 08/3/04
Another domain coloring of sin(z) http://en.wikipedia.org/ Hyperbolic Function (A) 08/3/04
Domain Coloring of sin(x), cos(x), tan(x) sin ( z ) = sin ( x +i y) cos( z) = cos( x+i y) tan( z ) = tan( x +i y ) http://en.wikipedia.org/ Hyperbolic Function (A) 08/3/04
Coordinates Changes y st = [ [] [ ][ ] s = t t x y x y = s t x + y = t ( x, y ) ( s, t ) s ][ ] [] π π s = cos ( 4 ) sin ( 4 ) x sin ( π4 ) cos( π4 ) y t (, ) x [ [] [ ][ ] x = y s ][ ] [] π π x = cos( 4 ) sin ( 4 ) s sin( π ) cos( π ) t y 4 4 s t + s + t = x s + t = y Hyperbolic Function (A) 3 08/3/04
Area:S Angle:A S = A () Angle A Area S Radius R Vector P = (x, y) st = t = = x y = s t x + y = t S = t s = y (, ds = s (x+ y) ds s x ) (x+ y) S = = ln(x+ y ) = ln(x± x ) s Hyperbolic Function (A) ds s 4 08/3/04
Area:S Angle:A S = A () S = ln (x+ y) = ln (x± x ) y Angle A Area S Radius R Vector P = (x, y) cosh A = x st = t x y = s x + y = t A = cosh x cosh A = A (e + e A ) = x x = u + u ( u = ea ) u x u + = 0 (, x ) u = x ± x e A A = ln( x ± x ) s Hyperbolic Function (A) S = A = ln( x± x ) 5 08/3/04
Area:S Angle:A S = A (3) Angle A Area S Radius R Vector P = (x, y) m= m=0 R = x y / A = tan y x R = x y / y x ( ) A = tan m= P( x, y) R A= S R Hyperbolic Function (A) 6 08/3/04
References [] http://en.wikipedia.org/ [] J. S. Walther, A Unified Algorithm for Elementary Functions [3] J. Calvert, http://mysite.du.edu/~jcalvert/math/hyperb.htm 08/3/04