Polynomial, trigonometric, exponential and fractal curves

[Loren Booda]

What other curves are there that cannot be described by the above? Are trigonometric functions actually a special case of exponentials with complex powers?

 

[hypermorphism]

Since the trigonometric functions can be written as combinations of sines and the sine function can be written as a complex exponential, we have exponential and logarithmic functions covering them and their inverses.
The Weierstrass curve and other functions represented by infinite series and solutions of differential forms give an infinitude of curves that cannot be described by finite means. Proprietary functions such as the Lambert W-function also abound. As another addition, there is also a vast jungle of nowhere analytic curves that cannot be described with any analytic function.

 

[tacman]

Polynomial curves are algebraic curves that can be described by a finite number of terms in a single variable, such as x^2 + 2x + 3. Trigonometric curves are curves that involve trigonometric functions, such as sine and cosine, and can be used to model periodic phenomena. Exponential curves are curves that involve exponential functions, such as y = e^x, and can be used to model growth or decay. Fractal curves are curves that exhibit self-similarity at different scales, such as the famous Mandelbrot set.

Other curves that cannot be described by the above include logarithmic curves, hyperbolic curves, and parametric curves. Logarithmic curves involve logarithmic functions, such as y = log(x), and can be used to model relationships between variables that are not linear. Hyperbolic curves involve hyperbolic functions, such as y = sinh(x), and can be used to model various physical phenomena, such as the shape of a hanging chain. Parametric curves are curves that are described by a set of parametric equations, such as the cycloid curve.

It is true that trigonometric functions can be expressed using complex exponential functions. This is known as Euler's formula, which states that e^(ix) = cos(x) + i*sin(x), where i is the imaginary unit. Therefore, trigonometric functions can be seen as a special case of exponential functions with complex powers. However, this does not diminish the importance and usefulness of trigonometric functions in mathematics and other fields.