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Complex gamma function
Complex gamma function
The gamma function is a mathematical function that extends the factorial function to real and complex numbers. The gamma function is defined as:
where z is a complex number with a positive real part, and the integral is taken over the positive real axis.
The complex gamma function is an extension of the gamma function to the complex plane. It is defined as:
where z is a complex number with a positive real part, and the integral is taken over the positive real axis.
One of the most famous approximations for the gamma function is the Stirling approximation, which is given by:
This formula provides an approximation for the natural logarithm of the gamma function. It is valid for large values of z and is often used to simplify calculations involving the gamma function.
The Stirling approximation is also used to derive other important formulas and approximations in mathematics and physics. For example, the approximation can be used to derive the Bohr model of the hydrogen atom in quantum mechanics.
Overall, the complex gamma function is a fascinating and important mathematical function with many properties and applications. The Stirling approximation is one of the most famous and useful approximations for the gamma function, and is widely used in many areas of mathematics and science.
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