hyp_myst

hyp_myst - x 2 n-1(2 n-1 which look similar to T 2 n cos x...

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A mathematical mystery. .. Math 8 2009, Chris Lyons Using the exponential function e x , we can create two simple functions: cosh x = 1 2 ( e x + e - x ) , sinh x = 1 2 ( e x - e - x ) . Here cosh x is called the hyperbolic cosine and sinh x is called the hyperbolic sine . Why do we use these names for these functions, when their definitions have nothing to do with sine or cosine? It’s because these functions are eerily similar to sine and cosine. Here’s a list of similarities: 1. We have the identity (cosh x ) 2 - (sinh x ) 2 = 1 , which is close to the familiar identity (cos x ) 2 + (sin x ) 2 = 1 . 2. There are addition laws cosh( x + y ) = cosh x cosh y + sinh x sinh y sinh( x + y ) = sinh x cosh y + cosh x sinh y, which are similar to the more familiar addition laws cos( x + y ) = cos x cos y - sin x sin y sin( x + y ) = sin x cos y + cos x sin y. 3. The derivatives are d dx cosh x = sinh x, d dx sinh x = cosh x, similar to d dx cos x = - sin x, d dx sin x = cos x. 4. The Taylor polynomials centered at 0 are given by ( T 2 n cosh)( x ) = ( T 2 n +1 cosh)( x ) = 1 + x 2 2! + x 4 4! + x 6 6! + · · · + x 2 n (2 n )! ( T 2 n - 1 sinh)( x ) = ( T 2 n sinh)( x ) = x + x 3 3! + x 5 5! + x 7 7! + · · ·
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Unformatted text preview: + x 2 n-1 (2 n-1)! , which look similar to ( T 2 n cos)( x ) = ( T 2 n +1 cos)( x ) = 1-x 2 2! + x 4 4!-x 6 6! + · · · + (-1) n x 2 n (2 n )! ( T 2 n-1 sin)( x ) = ( T 2 n sin)( x ) = x + x 3 3! + x 5 5!-x 7 7! + · · · + (-1) n-1 x 2 n-1 (2 n-1)! , So this is strange: except for some differences in signs at various places, the functions cosh and sinh seem to satisfy a lot of the same identities as cos and sin. Why should these functions, whose definitions have nothing to do with sine and cosine and whose graphs look nothing like sine and cosine, remind us so much of sine and cosine??? As we’ll see soon in Ma 1a, these mysterious similarities are no coincidence; in fact, they’re an indication that some deeper truth is hiding beneath the surface. .. 1...
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