# Hypergeometric Function around z=1/2

• betel
In summary, the conversation discusses the need for the behaviour of the hypergeometric function 2F1 near z=1/2, specifically with certain values for variables such as x, mu, and k. The first derivative at z=1/2 is also mentioned, with the possibility of a closed-form expression in terms of gamma functions. However, Taylor series is deemed the best approach, although it may involve lengthy and complex expressions. The need to compare with a different result is also mentioned.
betel
Hello,

for some calculation I need the behaviour of the hypergeometric function 2F1 near $$z=\tfrac{1}{2}$$. Specifically I need

$$_2 F_1(\mu,1-\mu,k,\tfrac{1}{2}+i x)$$

with $$x\in \mathbb{R}$$ near 0, and $$1/2\leq\mu\leq 2$$, $$1\leq k \in \mathbb{N}$$.

Differentiating around x=0 and writing the Taylor series gives a result, although very nasty and not really useful.

Does anybody know of an expansion around this point?
betel

Did you see the equation 73 here?

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

I have a feeling that there may be a closed-form expression for the first derivative at z=1/2 in terms of gamma functions as well, but I can't find it. Taylor series seems like the way to go. Should be nicely convergent.

P.S. Wolfram Alpha does not kick up any closed-form expressions. That means there probably isn't any. First derivative is proportional to $_2 F_1(1+\mu,2-\mu,k+1,\tfrac{1}{2})$ and that's as far as we can get.

Last edited:
That is what i did so far. there is a closed form for the derivative too.
The total expression then involves quite a few Gamma functions unfortunately always depending on half of the parameters, and it is quite lengthy and nasty.

If I take the limit in the differential equation I derived my hypergeometric solution from, I get a different behaviour not in terms of a power series.
But to be able to compare it to a different result I need to derive it from the H.geom.F in order to get the right coefficients.

## 1. What is a hypergeometric function around z=1/2?

A hypergeometric function around z=1/2 is a type of special mathematical function that is defined as a power series with a variable z, where the coefficients of the series are given by the ratio of two functions of z. It is used in many areas of mathematics, physics, and engineering to describe various phenomena.

## 2. How is a hypergeometric function around z=1/2 calculated?

The hypergeometric function around z=1/2 can be calculated using various methods, such as the power series method, the continued fraction method, or the integral representation method. The specific method used depends on the specific hypergeometric function and its properties.

## 3. What are the applications of hypergeometric function around z=1/2?

The hypergeometric function around z=1/2 has many applications in different fields, including probability theory, statistics, number theory, and quantum mechanics. It is also used in solving differential equations and in modeling complex systems.

## 4. What are the properties of a hypergeometric function around z=1/2?

Some of the properties of a hypergeometric function around z=1/2 include its analyticity, its singularity structure, its behavior under transformation of variables, and its asymptotic behavior. It also has various special cases and identities that are useful in solving mathematical problems.

## 5. Are there any real-life examples of hypergeometric function around z=1/2?

Yes, there are many real-life examples of hypergeometric function around z=1/2. One example is in the field of statistics, where it is used to calculate the probability of drawing a specific combination of cards from a deck. It is also used in analyzing the behavior of random walks and in determining the probability of success in certain gambling games.

• Calculus
Replies
2
Views
1K
• Calculus
Replies
1
Views
1K
• Calculus
Replies
4
Views
167
• Calculus
Replies
1
Views
276
• Calculus
Replies
5
Views
1K
• Calculus
Replies
10
Views
3K
• Calculus
Replies
2
Views
2K
• Calculus
Replies
3
Views
2K
• Special and General Relativity
Replies
5
Views
259
• Calculus
Replies
13
Views
1K