Ana sayfa Bilimsel Modüller SymPy ile Calculus hesaplamaları – Bölüm 4 (Gamma ve Faktöriyel fonksiyonları)

SymPy ile Calculus hesaplamaları – Bölüm 4 (Gamma ve Faktöriyel fonksiyonları)

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PAYLAŞ

Bu yazımızda, Gamma fonksiyonundan, ayrıca bu fonksiyonun tam sayılar için kısıtlanmış özel bir formu olan faktöriyel fonksiyonundan bahsedeceğim.

Gamma fonksiyonun en genel formülü şu şekildedir :

Burada dikkat edilmesi gereken nokta, z’nin karmaşık bir sayı olmasıdır ! Ve bu sayının Reel kısmı pozitif kabul edilir. Re(z) > 0  

Karmaşık sayı olması sizi yanıltmasın. Bilindiği gibi karmaşık sayılar, tam sayıları kapsadığı için bir karmaşık sayı alsak dahi, imajiner kısmı 0 yaparsak bunu tam sayı olarak kabul edebiliriz.

Bu fonksiyonun bir diğer güzelliği, rasyonel sayılarında hesaplayabiliyor oluşudur.

Alternatif gösterim olarak :

Bu fonksiyonlar, 0 ve negatif z sayıları dışında sırasıyla Euler ve Weierstrass tarafından tanımlanmıştır. γ ise  Euler-Mascheroni sabiti‘dir. Bu sabitin ilk 50 terimi,               0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 … şeklindedir.

Gamma fonksiyonunu grafiğe dökersek şayet, şu tablo ile karşılaşırız :
Daha fazla bilgi için, Kaynak linklerini inceleyebilirsiniz. Şimdi bu hesaplamaları Python’da nasıl yapabiliriz ona bakacağız.

Şimdi ise, gamma fonksiyonun özel bir hali olan faktöriyel fonksiyonuna gelelim.

Faktöriyel fonksiyonu:

Genel olarak, n! = (n-1)(n-2)…1 genel formülü ile sıralı çarpımlara verilen isimdir. Önceden belirttiğimiz gibi faktöriyel fonksiyonu, Gamma fonksiyonunun tam sayılar için özel bir halidir. Gamma(n) = (n-1)!’e denk düşer.

Yeni bir yazıda görüşmek üzere..  🙂

Kaynaklar:

http://mathworld.wolfram.com/GammaFunction.html

http://www.wiki-zero.net/index.php?q=aHR0cHM6Ly90ci53aWtpcGVkaWEub3JnL3dpa2kvR2FtYV9mb25rc2l5b251

http://docs.sympy.org/0.7.1/tutorial.html

http://mathworld.wolfram.com/IncompleteGammaFunction.html

http://mathworld.wolfram.com/IncompleteGammaFunction.html

http://mathworld.wolfram.com/Factorial.html

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