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Lecture15 - 13.6 Trigonometric and Hyperbolic Functions...

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13.6 Trigonometric and Hyperbolic Functions With the complex exponential function, trigonometric and hyperbolic func- tions may also be generalized to the complex numbers: for z C , we define cos z := 1 2 ( e iz + e iz ) , sin z := 1 2 i ( e iz e iz ) , (639) as well as cosh z := 1 2 ( e z + e z ) , sinh z := 1 2 ( e z e z ) . (640) Since the complex exponential is entire, so are the functions cos, sin, cosh, and sinh. We may also consider quotients such as tan z := sin z cos z , tanh z := sinh z cosh z . (641) The derivatives of these functions are given like for the real functions, namely by cos z = sin z, sin z = cos z, (642) cosh z = sinh z, sinh z = cosh z. (643) We determine the real and imaginary parts from the definitions: cos( x + iy ) = 1 2 ( e i ( x + iy ) + e i ( x + iy ) ) = 1 2 ( e ix e y + e ix e y ) (644) = 1 2 e y (cos x + i sin x ) + 1 2 e y (cos x i sin x ) (645) = cos x 1 2 ( e y + e y ) i sin x 1 2 ( e y e y ) (646) = cos x cosh y i sin x sinh y. (647) In the same way, we find sin( x + iy ) = sin x cosh y + i cos x sinh y, (648) cosh( x + iy ) = cosh x cos y + i sinh x sin y, (649) sinh( x + iy ) = sinh x cos y + i cosh x sin y. (650) 109
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Absolute values are given by | cos z | 2 = cos 2 x cosh 2 y + sin 2 x sinh 2 y (651) cosh 2 y sinh 2 y =1 = cos 2 x (1 + sinh 2 y ) + sin 2 x sinh 2 y (652) = cos 2 x + (cos 2 x + sin 2 x ) sinh 2 y (653) = cos 2 x + sinh 2 y, (654) and in the same way | sin z | 2 = sin 2 x + sinh 2 y (655) | cosh z | 2 = cosh 2 x sin 2 y (656) | sinh z | 2 = sinh 2 x + sin 2 y (657) Notice that the complex trigonometric functions sin, cos are not bounded.
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