Euler's Formula

A Euler's Formula is a mathematical equation (A=B) where A is a complex exponential function and B is the sum of trigonometric functions.

• Context
  • It can be defined as [math]\displaystyle{ e^{ix}=cosx + i sinx }[/math] , for any real number [math]\displaystyle{ x }[/math]
• Example(s):
  • [math]\displaystyle{ z=re^{i\theta}=r(cos\theta+isin\theta) }[/math]
  • [math]\displaystyle{ e^{i\pi}=cos(\pi)+isin(\pi)= -1 +i0= -1 }[/math]
  • [math]\displaystyle{ e^{i\pi/2}=cos(\pi/2)+isin(\pi/2)= 0+i1=i }[/math]
• Counter-Example(s):
  • [math]\displaystyle{ e^{2}= 7.3891 }[/math]
  • [math]\displaystyle{ |3+i4|=\sqrt{3^2+4^2}=5 }[/math]
• See: Complex Exponential Function, Euler Characteristic, Complex Number, Exponential Function, Imaginary Unit.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Euler's_formula Retrieved:2015-11-10.
  • Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number  : [math]\displaystyle{ e^{ix}=\cos x+i\sin x }[/math] where is the base of the natural logarithm, is the imaginary unit, and and are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted ("cosine plus i sine"). The formula is still valid if is a complex number, and so some authors refer to the more general complex version as Euler's formula. Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics."

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