متطابقة أويلر

للاستخدامات الأخرى، انظر قائمة المواضيع المسماة على اسم ليونارد أويلر#متطابقات أويلر

الدالة الأسية e^z can be defined as the limit of (1 + z/N)^N, as N approaches infinity, and thus e^iπ is the limit of (1 + iπ/N)^N. In this animation N takes various increasing values from 1 to 100. The computation of (1 + iπ/N)^N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 + iπ/N)^N. It can be seen that as N gets larger (1 + iπ/N)^N approaches a limit of −1.

In mathematics, Euler's identity (also known as Euler's equation) is the equality

$e^{i\pi }+1=0$

حيث

e is Euler's number, the base of natural logarithms,

i is the imaginary unit, which satisfies i² = −1, and

π is pi, the ratio of the circumference of a circle to its diameter.

Euler's identity is named after the Swiss mathematician Leonhard Euler. It is considered to be an example of mathematical beauty, perhaps a supreme example as it shows a profound connection between the most fundamental numbers in mathematics.

 شرح

Euler's formula for a general angle

Euler's identity is a special case of Euler's formula from complex analysis, which states that for any real number x,

$e^{ix}=\cos x+i\sin x$

where the inputs of the trigonometric functions sine and cosine are given in radians.

In particular, when x = π, or one half-turn (180°) around a circle:

$e^{i\pi }=\cos \pi +i\sin \pi .$

حيث

$\cos \pi =-1$

و

$\sin \pi =0,$

مما يستتبع أن

$e^{i\pi }=-1+0i,$

التي تنتج متطابقة أويلر:

$e^{i\pi }+1=0.$

 الجمال الرياضي

تشتهر متطابقة أويلر بشكل ملحوظ لجمالها الرياضي. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. كما تجمع هذه المتطابقة بين خمس من أهم الثوابت الرياضية:

• الصفر.
• الواحد.
• ط أو π.
• هـ أو e.
• العدد التخيلي i, the imaginary unit of the complex numbers.

 انظر أيضاً

• De Moivre's formula
• Exponential function
• Gelfond's constant

 المصادر

• Conway, John H., and Guy, Richard K. (1996). The Book of Numbers (Springer) ISBN 978-0-387-97993-9
• Crease, Robert P., "The greatest equations ever", Physics World, 10 May 2004 (registration required)
• Dunham, William (1999), Euler: The Master of Us All. Mathematical Association of America ISBN 978-0-88385-328-3
• Euler, Leonhard. Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus (Leipzig: B. G. Teubneri, 1922)
• Kasner, E., and Newman, J., Mathematics and the Imagination (Simon & Schuster, 1940)
• Maor, Eli, e: The Story of a number (Princeton University Press, 1998) ISBN 0-691-05854-7
• Nahin, Paul J., Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (Princeton University Press, 2006) ISBN 978-0-691-11822-2
• Paulos, John Allen, Beyond Numeracy: An Uncommon Dictionary of Mathematics (Penguin Books, 1992) ISBN 0-14-014574-5
• Reid, Constance, From Zero to Infinity (Mathematical Association of America, various editions)
• Sandifer, C. Edward. Euler's Greatest Hits (Mathematical Association of America, 2007) ISBN 978-0-88385-563-8
• Zeki, S.; Romaya, J. P.; Benincasa, D. M. T.; Atiyah, M. F. (2014), "The experience of mathematical beauty and its neural correlates", Frontiers in Human Neuroscience 8, doi:10.3389/fnhum.2014.00068 

 وصلات خارجية

• Complete derivation of Euler's identity
• Intuitive understanding of Euler's formula