# Relation between Beta and Gamma Function

## Introduction

In physics, the relation of” beta and gamma functions” is popular to compute the various different functions. Gama is a type of single-variable function that represents the variable functions of the particles while beta is a type of dual variable function that is applicable for computing as well as representing the amplitude in terms of reggae trajectories.

The relationship between the beta and gamma function is presented with the formula

“B(p,q)=(Tp. Tq)/T(p+q)”

In order to evaluate the integral functions of the gamma beta function, this formula is very crucial.

## What is the relation between beta and gamma function?

Leonhard Euler first introduced the function of “beta and gamma” and this function is mostly used for providing the extensions of the particle distributions in the field of “Gauss hyper geometric functions” along with  Confluent hyper geometric function.

This function is also applicable for proving the extension of the derivatives of “Riemann-Liouville” (Nascimento et al. 2019).

Figure 1: The gamma function for calculating the real number

In physics, gamma functions were developed for dealing with the interpolation problem. In terms of thermal expansion, the “beta and gamma functions” are hugely applicable. The beta function is severely used in extracting the results from the “Coefficient of areal expansion” while the function of gamma is applicable for the “Coefficient of volumetric expansion”. Besides this, in order to detect the range of temperature of the particles these two functions are applied. This is an important type of function that is also popular as the primary type of Euler’s integrals (May et al. 2019).

β is the notation that is mostly used to denote the beta functions. In physics, beta is also denoted by the p and q, in which both values denote real values.

## Difference between beta and gamma function

Beta function Gamma function
• Beta function is a double integral function.
• The function is denoted with m and n.
• The beta function is quite simpler compared to the gamma and alpha functions. Additionally, this function is used for simplifying the gamma functions.
• Beta function is mostly applicable for different purposes because it is very handy compared to other functions.
• The application of the beta function is commonly seen in the “Stochastic Urn Process”
• The gamma function is denoted with a single integral function.
• This function is denoted with only n.
• This function relates to a complex structure of physics.
• The use of the gamma function is not applicable to detect the preferential attachment of the particles as it is complex in form.
• The use of gamma function is commonly seen in the process of preferential attachment.

Table 1: Difference between “beta and gamma function”

## The properties of “beta and gamma function”

Figure2: Gamma function solutions

In the process of the stochastic urn the application of beta function guides to detect the meaning of the process that discrete the wealth units that are commonly known as balls.

Here, S1 and S2 are the initial and the final speed of the particles that can be measured through the application of “beta and gamma functions” (Sun et al. 2018).

For processing the random as well as partially random application of an object the beta function is also used The process of the stochastic urn is also comprised of the process of A preferential attachment, in which the beta functions distribute the balls among the urns to increase the number of functions (Douma & Weedon, 2019). While the function of gamma is mostly applicable to reduce the number of functions as the balls already have the number of functions.

## Application of beta and gamma function

Figure 3: The relation between beta and gamma functions

The application of the beta function in physics is mostly seen in the String theory as is primarily known as “Scattering amplitude”. Recently in physics, the beta functions are also applicable in the “Stochastic Urn Process”. As the number of urns increases rapidly then the beta functions are applicable for reducing the flow (Maia & Sousa, 2019). The Gamma function is a complex integral function, so the beta function is used to simplify the complexity of the gamma functions.

## Conclusion

The relationship between “beta and gamma functions” varies in the inputs and output functions of the particles. The functions of beta are highly associated with each input as well as the output value. The beta function is expressed as "β" which is denoted by x and y. In the beta functions, the values of x and y are real as well as sometimes greater than “0”. The string theory is a complex part of physics the beta functions represents the amplitudes in terms of Regge trajectories. In theoretical physics particular quantum theory, this function encodes the reliance of coupling parameters on an energy scale.

## FAQs

Q.1 Is the gamma function called the Euler’s integral?

The functions of gamma are also called a second king of the Euler’s integral. This function is also connected with the detection of the wealth of the particles.

Q.2 What are the properties of beta and gamma function?

In the making of MATLAB software, the beta and gamma solutions are severally applicable. The application of the Riemann-Liouville fractional derivative operator through the relationship of these two particles is mostly used to detect the functions of image formulas.

Q.3. What is the use of beta and gamma functions?

The relationship between the gamma function is mostly applicable as the beta function is dependent on the two different variables m and m. The gamma function entirely relies on the single variable m.

Q.4 How the relationship between the beta and gamma functions are measured?

The relationship of the these two functions can be presented as β (m,n) = ΓmΓn/ Γ (m+n) Where, β (m,n). In this equation, the relationship between the function is entirely dependent on the two most important variables m, and n.

## References

### Journals

Douma, J. C., & Weedon, J. T. (2019). Analysing continuous proportions in ecology and evolution: A practical introduction to beta and Dirichlet regression. Methods in Ecology and Evolution, 10(9), 1412-1430. Retrieved from: https://besjournals.onlinelibrary.wiley.com/doi/pdf/10.1111/2041-210X.13234

Maia, M. A., & Sousa, E. (2019). BACE-1 and γ-secretase as therapeutic targets for Alzheimer’s disease. Pharmaceuticals, 12(1), 41. Retrieved from: https://www.mdpi.com/1424-8247/12/1/41/pdf

May, E. S., Nickel, M. M., Ta Dinh, S., Tiemann, L., Heitmann, H., Voth, I., ... & Ploner, M. (2019). Prefrontal gamma oscillations reflect ongoing pain intensity in chronic back pain patients. Human brain mapping, 40(1), 293-305. Retrieved from: https://onlinelibrary.wiley.com/doi/pdf/10.1002/hbm.24373

Nascimento, A. D., Silva, K. F., Cordeiro, G. M., Alizadeh, M., Yousof, H. M., & Hamedani, G. G. (2019). The odd Nadarajah-Haghighi family of distributions: properties and applications. Studia Scientiarum Mathematicarum Hungarica, 56(2), 185-210. Retrieved from: https://epublications.marquette.edu

Sun, B., Yan, M., Feng, Q., Li, Y., Ren, Y., Zhou, K., & Zhang, W. (2018). Gamma degradation process and accelerated model combined reliability analysis method for rubber O-rings. IEEE Access, 6, 10581-10590. Retrieved from: https://ieeexplore.ieee.org/iel7/6287639/6514899/08275508.pdf

### Websites

curiophysics (2022), the relationship between the beta and gamma functions , Available at: https://curiophysics.com/ [Accessed on 11th June 2022]

Updated on: 13-Oct-2022

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