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sin hx, cos hx, tan h x.... What are they? What does the "h" mean here?
Any explanations or websites would be appreciated.
Any explanations or websites would be appreciated.
I guess z is a complex number, right?Originally posted by Lonewolf
Also:
sinh x = -i*sin(i*z)
cosh x = cos(i*z)
Where i is sqrt(-1)
Differentiate sin hx = e^{x} (interesting )sinh x = -i*sin(i*z) ...........(1)
sinh x = (e^{x}-e^{-x})/2...........(2)
Lonewolf meant to use x on both sides, I think.Originally posted by KL Kam
I guess z is a complex number, right?
No, you get:Differentiate sin hx = e^{x} (interesting )
Yes, the above holds for complex variables, too.but I don't think a function involving complex number, -i*sin(i*z), is differentiable, or am I wrong?
Again, I think Lonewolf meant to put x on both sides.Why -i*sin(i*z) = e^{x}-e^{-x})/2 = sin hx ? [/B]
#Lonewolf meant to use x on both sides, I think.
oops, I swollowed the "-" sign when I differentiated e^(-x)d(sinh x)/dx=(1/2)(e^{x}+e^{-x})
Is it because we can represent a function with complex variables by using an Argand(sp?) Diagram, just as we use a Cartesian plane to represent functions with real variables?Yes, you can differentiate functions with a complex variable, in almost the same way as a real function.
It is because the "limit" definition of a derivative holds equally well for complex numbers.Originally posted by KL Kam
Is it because we can represent a function with complex variables by using an Argand(sp?) Diagram, just as we use a Cartesian plane to represent functions with real variables?
Yes.(1/2)(e^{x}+e^{-x})=cosh x
Is it the defination of cos hx ?
There is an analog. You can invert the system of equations:I've heard of Euler's formula e^{ix} = cos x + i sin x
are the hyperbolic functions somehow related to Euler's formula?
X = A sin h θ
In almost exactly the same way. We use the inverse function sinh^{-1}θhow can we write θ in terms of X and A ?