Beta Fonksiyonu için Sayısal Değerlendirme Yöntemleri

Abstract

Bu çalışmada, hesaplamalı matematik ve fizikte karşılaşılan beta fonksiyonu analiz edilmiştir. Bu fonksiyonun doğru değerlendirilmesi, kuantum mekaniksel hesaplamalardaki diğer matematiksel fonksiyonların doğruluğunu da etkilemektedir. Özellikle son yıllarda, sıfır ve negatif p ve q tam sayıları için beta fonksiyonu ile ilgili çalışmalara ilgi duyulmaktadır. Bu çalışma, beta fonksiyonunun neutrix limitlerini göz önünde bulundurarak, özellikle negatif p ve q tam sayılarında beta fonksiyonunun sayısal hesaplaması için yeni bağıntılar sunmaktadır. Ayrıca pozitif p ve q tam sayı değerleri için de beta fonksiyonunun tanımı dikkate alınarak, fonksiyonun tüm tam sayı değerlerinde hesaplanması için bir algoritma oluşturulmuştur. Son olarak, yeni yineleme bağıntılarımız ve algoritmamız yardımıyla elde edilen sayısal sonuçlar sunulmuştur.

Keywords

• Beta fonksiyonu
• Gama fonksiyonu
• Kuantum sayıları
• Mathematica

The Numerical Evaluation Methods for Beta Function

Abstract

In this study, the beta function that is encountered in computational mathematics and physics is analyzed. The correct evaluation of this function also affects the accuracy of other mathematical functions in quantum mechanical calculations. Especially in recent years, there is an interest in studies related to the beta function for zero and negative p and q integers. This study, considering the neutrix limits of the beta function, presents new relations for the numerical computation of the beta function, especially for negative integers p and q. In addition, taking into account the definition of the beta function for positive p and q integer values, an algorithm is created to calculate the function for all integer values. Finally, numerical results obtained with the help of our new recurrence relations and algorithm are presented.

Keywords

• Beta function
• Gamma function
• Quantum numbers
• Mathematica

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