Polynomial, trigonometric, exponential and fractal curves

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SUMMARY

This discussion explores the relationship between various types of curves, specifically polynomial, trigonometric, exponential, and fractal curves. It establishes that trigonometric functions can be expressed as complex exponentials, highlighting their connection to exponential and logarithmic functions. Additionally, it introduces the Weierstrass curve and other infinite series functions, emphasizing the existence of numerous curves that cannot be represented by finite means. Proprietary functions like the Lambert W-function and non-analytic curves are also noted as significant examples of this diversity.

PREREQUISITES
  • Understanding of complex exponentials and their relation to trigonometric functions
  • Familiarity with infinite series and differential equations
  • Knowledge of the Weierstrass curve and its properties
  • Awareness of proprietary functions such as the Lambert W-function
NEXT STEPS
  • Research the properties and applications of the Weierstrass curve
  • Study the Lambert W-function and its implications in complex analysis
  • Explore the concept of nowhere analytic curves and their characteristics
  • Investigate the relationship between trigonometric functions and complex exponentials in depth
USEFUL FOR

Mathematicians, physicists, and computer scientists interested in advanced curve analysis, complex analysis, and the study of non-analytic functions.

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