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Euler's formula, named after Leonhard Euler , is a Mathematical formula in Complex Analysis that shows a deep relationship between the Trigonometric Functions and the Complex Exponential Function . ( Euler's Identity is a special case of the Euler formula.) Euler's formula states that, for any Real Number ''x'', : where : and are Trigonometric Function s. Richard Feynman called Euler's formula "our jewel" and "the most remarkable formula in mathematics" (Feynman, p. 22-10). HISTORY Euler's formula was proved (in an obscured form) for the first time by ). APPLICATIONS IN COMPLEX NUMBER THEORY This formula can be interpreted as saying that the function ''e''''ix'' traces out the unit circle in the Complex Number Plane as ''x'' ranges through the real numbers. Here, ''x'' is the Angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in radians. The formula is valid only if sin and cos take their arguments in radians rather than in degrees. The proof is based on the Taylor Series expansions of the Exponential Function ''e''''z'' (where ''z'' is a complex number) and of sin ''x'' and cos ''x'' for real numbers ''x'' (see below). In fact, the same proof shows that Euler's formula is even valid for all ''complex'' numbers ''x''. Euler's formula can be used to represent complex numbers in Polar Coordinates . Any complex number ''z''=''x''+''iy'' can be written as : where : : |
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