- #1
DMESONS
- 27
- 0
Can someone help me to expand the following gamma functions around the pole ε, at fisrt order in ε
[itex]\Gamma[(1/2) \pm (ε/2)][/itex]
where ε= d-4
[itex]\Gamma[(1/2) \pm (ε/2)][/itex]
where ε= d-4
Expanding Gamma function around poles
- Thread starter DMESONS
- Start date
In summary, the conversation is about expanding gamma functions around the pole ε at first order in ε, and the use of the digamma function and values for Γ(½) and ψ(½). The conversation also includes a thank you to Bill_K and Hepth for their help and a mention of being new to the subject.
- #2
Bill_K
Science Advisor
- 4,158
- 205
Γ(½ ± ε/2) ≈ Γ(½) ± ε/2 Γ'(½)
No, seriously..
Well, you also need to use the digamma function, ψ(x) = Γ'(x)/Γ(x). And the values Γ(½) = √π and ψ(½) = - γ - 2 ln 2 where γ is Euler's constant.
No, seriously..
Well, you also need to use the digamma function, ψ(x) = Γ'(x)/Γ(x). And the values Γ(½) = √π and ψ(½) = - γ - 2 ln 2 where γ is Euler's constant.
- #3
Hepth
Gold Member
- 464
- 40
[tex]
\Gamma(\frac{1}{2} - \frac{\epsilon}{2}) = \sqrt{\pi }+\frac{1}{2} \sqrt{\pi } \epsilon (\gamma_E +\log (4))+O\left(\epsilon ^2\right)
[/tex]
[tex]
\Gamma(\frac{1}{2} + \frac{\epsilon}{2}) = \sqrt{\pi }+\frac{\sqrt{\pi } \epsilon \psi ^{(0)}\left(\frac{1}{2}\right)}{2}+O\left(\epsilon ^2\right)
[/tex]
\Gamma(\frac{1}{2} - \frac{\epsilon}{2}) = \sqrt{\pi }+\frac{1}{2} \sqrt{\pi } \epsilon (\gamma_E +\log (4))+O\left(\epsilon ^2\right)
[/tex]
[tex]
\Gamma(\frac{1}{2} + \frac{\epsilon}{2}) = \sqrt{\pi }+\frac{\sqrt{\pi } \epsilon \psi ^{(0)}\left(\frac{1}{2}\right)}{2}+O\left(\epsilon ^2\right)
[/tex]
- #4
DMESONS
- 27
- 0
Bill_K and Hepth, I am so grateful for your help
I am new in this subject
I am new in this subject
Related to Expanding Gamma function around poles
What is the Gamma function?
The Gamma function is a mathematical function used in many areas of mathematics, including statistics, number theory, and physics. It is defined as an extension of the factorial function to complex and real numbers.
What are poles in the Gamma function?
Poles in the Gamma function are points where the function is undefined or infinite. These points are typically located at negative integers and zero, and they play a crucial role in the expansion of the Gamma function.
Why is it important to expand the Gamma function around poles?
Expanding the Gamma function around poles allows for the evaluation of the function at these points, which would otherwise be undefined. This expansion is also useful in simplifying complex expressions involving the Gamma function.
How is the Gamma function expanded around poles?
The Gamma function can be expanded around poles using the Laurent series, which is a representation of a function as a sum of infinitely many terms. This series involves the use of negative powers of the variable, making it suitable for expanding around poles.
What are some applications of expanding the Gamma function around poles?
Expanding the Gamma function around poles has many applications, including in the evaluation of complex integrals, solving differential equations, and in the study of special functions in mathematics and physics.
Similar threads
-
Calculus
-
Calculus
-
Calculus
-
Classical Physics
-
Calculus
-
Advanced Physics Homework Help