Complex Trigonometric and Hyperbolic Functions (7A)

Complex Trigonometric and Hyperbolic Functions (7A) 07/08/015

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cosh(x) 1 +α α cosh α = (e + e ) cos θ = 1 + jθ (e + e j θ ) http://en.wikipedia.org/ 3 07/08/015

sinh(x) sinh α = sin θ = 1 +α α (e e ) 1 + jθ (e e j θ ) j http://en.wikipedia.org/ 4 07/08/015

Definitions of Hyperbolic Functions cosh = 1 e e sinh = 1 e e e e tanh = e e http://en.wikipedia.org/ 5 07/08/015

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 = cosh x sinh x cos x = 1 +ix (e + e ix ) cosh x = (e + e ) sin x = 1 +ix (e e ix ) i sinh x = (e e x ) + ix ix 1 (e e ) tan x = i (e+ix + e ix ) (e+ x e ) tanh x = (e+ x + e ) 6 07/08/015

Trigonometric functions with imaginary arguments ix x cos x = 1 +ix (e + e ix ) cosh x = (e + e ) sin x = 1 +ix (e e ix ) i sinh x = (e e ) + ix ix 1 (e e ) tan x = i (e+ix + e ix ) (e+ x e ) tanh x = (e+ x + e ) cos i x = cosh x cos i x = 1 +x (e + e ) cosh x = (e + e ) sin i x = i sinh x sin i x = 1 (e e + x ) i sinh x = (e e ) tan i x = i tanh x +x 1 (e e ) tan i x = i (e x + e + x ) 7 (e+ x e ) tanh x = (e+ x + e ) 07/08/015

Hyperbolic functions with imaginary arguments cos x = 1 +ix (e + e ix ) cosh x = (e + e ) sin x = 1 +ix (e e ix ) i sinh x = (e e ) + ix ix 1 (e e ) tan x = i (e+ix + e ix ) x ix (e+ x e ) tanh x = (e+ x + e ) 1 cos x = (e +ix + e ix ) cosh i x = 1 +i x i x (e + e ) cosh i x = cos x 1 +ix ix sin x = (e e ) i sinh i x = 1 +i x (e e i x ) sinh i x = i sin x + ix ix 1 (e e ) tan x = i (e+ix + e ix ) (e+i x e i x ) tanh i x = (e +i x + e i x ) 8 tanh i x = i tan x 07/08/015

With imaginary arguments cos x = 1 +ix (e + e ix ) cosh x = (e + e ) sin x = 1 +ix (e e ix ) i sinh x = (e e ) + ix ix 1 (e e ) tan x = i (e+ix + e ix ) (e+ x e ) tanh x = (e+ x + e ) 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 x 9 ix 07/08/015

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 10 07/08/015

Modulus of sin(z) (1) 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) = 1 i = 1 4 (e y +i x e+ y i x )(e y i x e + y +i x ) = 1 4 (e y e+ i x e i x + e+ y ) = 1 4 (e+ y + e y e + i x + e i x ) (e +i (x +i y ) e i( x +i y ) 1 i ) (e i (x i y) e + i( x i y ) ) = + 14 (e+ y + e y ) 14 (e+ i x + e i x ) = [ 1 (e + y e y ) ] + [ 1i (e+i x e i x ) ] = sin (x ) + sinh ( y ) 11 07/08/015

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 = 1 = sin (x )(1 + sinh ( y )) + (1 sin ( x))sinh ( y) cos α + sin α = 1 = sin (x ) + sin (x)sinh ( y) + sinh ( y ) sin ( x)sinh ( y ) = sin (x ) + sinh ( y ) 1 07/08/015

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/ 13 07/08/015

Domain Coloring Argument Domain z Coloring f ( z) Argument of f(z) http://en.wikipedia.org/ 14 07/08/015

Domain Coloring Modulus Domain z Coloring f ( z) Modulus of f(z) http://en.wikipedia.org/ 15 07/08/015

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/ 16 07/08/015

Another domain coloring of sin(z) http://en.wikipedia.org/ 17 07/08/015

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/ 18 07/08/015

References [1] http://en.wikipedia.org/ [] J. S. Walther, A Unified Algorithm for Elementary Functions [3] J. Calvert, http://mysite.du.edu/~jcalvert/math/hyperb.htm 07/08/015