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Re(cos(z)) on [-τ,τ]²
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The real component of cosine on the complex plane, with a path traced from -iÏ€ (-2Ï€) to iÏ€ (2Ï€) along each imaginary axis, illustrates how slicing at Im(z) = 0 yields the conventional cosine function, while slices at Im(z) = ï„n (n ∈ â„•) display hyperbolic cosine (cosh). This concept stems from Euler's formula, where cos(x) = 1/2(e^(iθ) + e^(-iθ)), as the imaginary parts of two exponentials cancel each other out for real components, resulting in twice the real component. In a complex scenario, increasing the imaginary part alters the radius of the circle one is traversing while reducing that of the other, leading to non-canceling imaginary components and wider waves. This model was developed using precursors to surfcad software; further insights can be found at https://christopherolah.wordpress.com/2011/07/16/surface-oriented-cad-math-telescopes/.
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