متطابقة أويلر

	جزء من سلسلة منطقات عن; الثابت الرياضي، e
	لوغاريتم طبيعي
	التطبيقات في: الفائدة المركبة • متطابقة اويلر وصيغة اويلر • نصف العمر والنمو/الانحلال الأسي
	تعريف e: إثبات حتى e غير كسرية • تمثيلات e • مبرهنة ليندمان–ڤايرشتراس
	شخصيات جون ناپييه • ليونهارد اويلر;
	حدسية شانويل;

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 i π

is the limit of

(1 + i π / N) N

. In this animation

N

takes various increasing values from 1 to 100. The computation of

(1 + i π / 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 + i π / N) N

. It can be seen that as

N

gets larger

(1 + i π / N) N

approaches a limit of −1.

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

{\displaystyle e^{i\pi +1=0

حيث

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

$i$ is the imaginary unit, which satisfies

i 2 = -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 $x$ ,

{\displaystyle 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:

{\displaystyle e^{i\pi =\cos \pi +i\sin \pi .

حيث

{\displaystyle \cos \pi =-1

و

{\displaystyle \sin \pi =0,

مما يستتبع أن

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

التي تنتج متطابقة أويلر:

{\displaystyle 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 أبرز الثوابت الرياضية:

الصفر.
الواحد.
ط أوπ.
هـ أوe.
العدد التخيلي i, the imaginary unit of the complex numbers.

انظر أيضاً

De Moivre's formula
Exponential function
Gelfond's constant

ملاحظات

^ The term "Euler's identity" (or "Euler identity") is also used elsewhere to refer to other concepts, including the related general formula $e ix = cos x + i sin x$ , and the Euler product formula.

المراجع

^ Dunham, 1999, p. xxiv.
^ Stepanov, S. A. (7 February 2011). "Euler identity". Encyclopedia of Mathematics. Retrieved 18 February 2014.
^ Gallagher, James (13 February 2014). "Mathematics: Why the brain sees maths as beauty". BBC News Online. Retrieved 26 December 2017. Italic or bold markup not allowed in: |website= (help)

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