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= + , where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions.
Uass ae ix. è ÷\Õ 1\¹ · °\Ñ\Ù\ k 0 ±\Ò È\¹\Ô\Í\Ð\®\õ Ã ¼\¶ W\¹\ \½\Ø\½\Ò\Ò ê è9\ ê è û \Ø 8 u ?. ` o M b. 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 x:.
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{ Þ ª » q Ô w Ý ¯ t ª » Ì U s Ô ù U K b w p z è q ` o S ß Q X i ^ M {1 í K h ÿ1 Ì ¡!!. ï í l í ñ l î ì í ò ð X ó ^ u µ v P ' o Æ Ç :. P _ ` & d c d + c ß à Î 3 ¦ r x Ø Ì 9 Þ 3 ¦ < i ß à < Ü Ï 3 ¦ u ó ¤ Ú d È Ï f ² g f a g f t g f \ g Ü y ² + a t # \ È õ 6.
ó ò l í l î ì í ñ ñ X ï ^ u µ v P ^ ñ ð l í í l î ì í ð ò X ð. ^ u µ v P ' o Æ Ç :. E i x = cos( x ) + i sin( x ) Justification #1:.
From the derivative Consider the function on the right hand side (RHS) f(x) = cos( x ) + i sin( x ) Differentiate this function f ' (x) = -sin( x ) + i cos( x) = i f(x) So, this function has the property that its derivative is i times the original function. \Ò\Ø Ý ¬\¶ å\ó\µ\Ò\Ô\Í\Ð\®\è\Ã\. ö 6 J Ã r.
< É æ ß ¥ 2 ® Æ ¥ ¬ æ L < ® Å N w q Q x í Û > • / H É ï × ¥ Ó É a í ã å < ' õ J = Â µ ´ æ Þ ß ï Æ Ý Å å ½ æ > – £ Ë â > Æ Ð ~ ² º L < É ö E I ¨ Ã ¾ º Õ µ É X Ð í Ö ¼ ó É { Ü ¦ ä ¸ ² º å ) ;. ª » Ì x 1. What other type of function has this.
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