What Is the Gamma Function

The gamma function is a somewhat complicated function.   This function is used in mathematical statistics.   It can be thought of as a way to generalize the factorial.   The Factorial as a Function We learn fairly early in our mathematics career that the factorial, defined for non-negative integers n, is a way to describe repeated multiplication. It is denoted by the use of an exclamation mark. For example:​ 3! 3 x 2 x 1 6 and 5! 5 x 4 x 3 x 2 x 1 120. The one exception to this definition is zero factorial, where 0! 1. As we look at these values for the factorial, we could pair n with n!. This would give us the points (0, 1), (1, 1), (2, 2), (3, 6), (4, 24), (5, 120), (6, 720), and so on. If we plot these points, we may ask a few questions: Is there a way to connect the dots and fill in the graph for more values?Is there a function that matches the factorial for nonnegative whole numbers, but is defined on a larger subset of the real numbers. The answer to these questions is, â€Å"The gamma function.† Definition of the Gamma Function The definition of the gamma function is very complex. It involves a complicated looking formula that looks very strange. The gamma function uses some calculus in its definition, as well as the number e Unlike more familiar functions such as polynomials or trigonometric functions, the gamma function is defined as the improper integral of another function. The gamma function is denoted by a capital letter gamma from the Greek alphabet. This looks like the following: Γ( z ) Features of the Gamma Function The definition of the gamma function can be used to demonstrate a number of identities. One of the most important of these is that Γ( z 1 ) z Γ( z ). We can use this, and the fact that Γ( 1 ) 1 from the direct calculation: Γ( n ) (n - 1) Γ( n - 1 ) (n - 1) (n - 2) Γ( n - 2 ) (n - 1)! The above formula establishes the connection between the factorial and the gamma function. It also gives us another reason why it makes sense to define the value of zero factorial to be equal to 1. But we need not enter only whole numbers into the gamma function. Any complex number that is not a negative integer is in the domain of the gamma function. This means that we can extend the factorial to numbers other than nonnegative integers. Of these values, one of the most well known (and surprising) results is that Γ( 1/2 ) √π. Another result that is similar to the last one is that Γ( 1/2 ) -2Ï€. Indeed, the gamma function always produces an output of a multiple of the square root of pi when an odd multiple of 1/2 is input into the function. Use of the Gamma Function The gamma function shows up in many, seemingly unrelated, fields of mathematics. In particular, the generalization of the factorial provided by the gamma function is helpful in some combinatorics and probability problems. Some probability distributions are defined directly in terms of the gamma function. For example, the gamma distribution is stated in terms of the gamma function. This distribution can be used to model the interval of time between earthquakes. Students t distribution, which can be used for data where we have an unknown population standard deviation, and the chi-square distribution are also defined in terms of the gamma function.

The Trial And Death Of Socrates - 1701 Words

In reading, The Trial and Death of Socrates, Socrates presents himself against â€Å"the majority† and endures prosecution for inventing new gods and not believing in old ones. Socrates lives a life that is opposite of much of the â€Å"the majority† ideals. â€Å"The majority† values goods, possessions, and uncontrollable attributes given to them by others such as a high reputation and honor. In addition, The Greek Philosopher was able to refuse the opportunity to escape his death sentence while he had sons at home because he does not see death as the greatest evil and provided his sons a positive role model. He showed calmness in the face of his trouble, while the vulnerable and fragile majority would have been in constant fear of death. Socrates, in saying goodbye to Crito in his cell â€Å"Then let it be, Crito, and let us do as I say, seeing that God so directs us.† (Plato, 102) Socrates believes that you cannot harm a good man in life or death. Through The Trial and Death of Socrates, Socrates’ has to overcome a test of his ideals. He proves to the people and the readers that he is unharmable. If a person suffers a loss of a valued good of life, it causes the person to be harmed due to the loss or absence disrupting the status of their happiness. The the loss of goods by forces beyond their control can afflict an individual. Socrates knew â€Å"the majority† recognized wealth, honor, reputation, possessions, and power as goods of life. An ordinary man would fear the loss of those, as theyShow MoreRelatedThe Trial And Death Of Socrates : The Trial And Death Of Socrates972 Words   |  4 Pages The Trial and Death of Socrates, depicts the different stages of Socrates’s life, from his prosecution until his execution. During the narratives, Socrates gives us, as readers, insight towards his beliefs and philosophy, which are viewed as reas ons for his imprisonment. Phaedo’s recollection of Socrates’s last few hours alive, reveals Socrates’s most important belief, that the soul is an entity which is immortal and is valuable during and after life. All perceived lusts, greed, and fear are causedRead MoreThe Trial And Death Of Socrates845 Words   |  4 Pagesright or not. For that, in this essay, we will use Plato’s The Trial and Death of Socrates, as our example of how this kind of attitudes play in Socrates’ life as it is well-known as intelligently and morally lived one. In Euthyphro section, Socrates meets a man named Euthyphro before his trial, where he is being accused by Meletus for corrupting the youth. While Euthyphro is here to prosecute his own father for the case of murder. Socrates starts to ask him about his meaning of piety. â€Å"It is notRead MoreTrial and Death of Socrates2267 Words   |  10 PagesBenjamin Jowett. The Trial and Death of Socrates (Dover Edition). New York: Dover Publications, 1992 â€Å"What is the charge? Well, a very serious charge, which shows a good deal of character in the young man, and for which he is certainly not to be despised. He says he knows how the youth are corrupted and who are their corruptors. And I fancy that he must be a wise man, and seeing that I am anything but a wise man, he has found me out, and is going to accuse me of corrupting his young friendsRead MoreThe Trial and Death of Socrates Essay1639 Words   |  7 PagesThe portrayal of Socrates, through the book â€Å"the trial and death of Socrates† is one that has created a fairly controversial character in Western history. In many ways, Socrates changed the idea of common philosophy in ancient Greece; he transformed their view on philosophy from a study of why the way things are, into a consideration man. Specifically, he analyzed the virtue and health of the human soul. Along side commending Socrates for his strong beliefs, and having the courage to stand by thoseRead M oreThe Trial And Death Of Socrates And The Gita1563 Words   |  7 Pagesalso true of the ancient texts The Trial and Death of Socrates and The Gita, despite their conflicting messages on the importance of individuality. In The Trial and Death of Socrates, Socrates seems to take an individualist approach in that people choose to follow their â€Å"god†, or conscience, while The Gita argues that people are being guided by truth. The Gita shows this with Arjuna’s devotion to Krishna’s wishes despite his inherent ties to his family. Both Socrates and Arjuna find themselves at oddsRead MorePlato : The Trial And Death Of Socrates1107 Words   |  5 PagesAmanda Kewal Phil 103 Essay #1 Plato: The Trial and Death of Socrates Euthyphro In Platos Euthyphro, Euthyphro is surprised to see Socrates at the courthouse because he doesnt seem like the kind of person that should be anywhere around a courthouse. They both discuss why Socrates is there in which he tells Euthyphro that he is being indicted by Meletus for corrupting the youth because he does not believe in the old gods of Athens and makes up new gods. Socrates goes on by telling Euthyphro his issueRead MorePlato: The Trial and Death of Socrates1263 Words   |  5 PagesPLATO- The Trial and Death of Socrates Athens belonged to a royal blood of an Aristocratic family. Plato was a good learner and pupil who always wanted to wanted to justify Socrates and tried to discover the eternal principles of human conduct, happiness, justice, temperance and courage. He was holding a unique and valuable position in the area when he was only about twenty eight years old. Plato never forced anyone to follow him and to adopt his thoughts in order to adopt a true and simple meanRead MorePlato s The Trial And Death Of Socrates Essay1671 Words   |  7 PagesPlato’s The Trial and Death of Socrates presents the reader with complex competing conceptions of what should be considered â€Å"the good life†. According to Socrates, â€Å"the most important thing is not life, but the good life† (Crito, 48b). The majority, who live a non-philosophical life, believes the goods of life include wealth, reputation, and honor: all things that can easily be taken away or destroyed. On the other hand, Socrates lives a philosophical life filled with self-sufficiency. He views wisdomRead MoreSocratic Justice And The Trial And Death Of Socrates992 Words   |  4 Pagesin The Trial and Death of Socrates. I will also go into detail about the differences between Machiavelli’s views of government as opposed to Socrates’s. Socrates and Machiavelli are two the greatest philosophers known today. Both had an effective way of showing and standing up for what they believe in. While both men were very intelligent, Socrates and Machiavelli differ with their views on justice. Socrates defines justice as doing what is best for the wellbeing of his country. Socrates shows thisRead MoreThe Trial and Untimely Death of Socrates Essay527 Words   |  3 PagesThe Trial and untimely death of Socrates, in my opinion, was a small group of people throwing a fit when it was pointed they weren’t as smart as they thought. The reasoning used by Socrates is the greatest example of the facts, not the manipulation of, proving your innocence. Socrates makes several points as to the trial being a complete waste of time and that even if he was brought to court he would be innocent. I agree death was the wrong verdict by the jury, the jury should have voted for innocence

Hate vs Fear Free Essays

Hate Vs Fear: â€Å"Fearing the unknown† Have you ever heard someone say â€Å"I hate this† or â€Å"I hate that†? Do you think they really hate that object, person or idea? Are they confusing hatred with fear? What exactly is fear? Fear is a distressing emotion aroused by impending danger, evil or pain, Whether or not the threat is real or imagined. What exactly is hate? Hate is to dislike intensely or passionately; feeling extreme aversion or extreme hostility toward; detest. Why is hate accepted in today’s culture? Saying a person hates something, someone or an idea is a lot easier for most people in today’s society, rather than saying they fear it. We will write a custom essay sample on Hate vs Fear or any similar topic only for you Order Now They do not have to explain themselves why exactly they â€Å"hate† something. It just is. We are taught from a young age to feel hostility or hatred towards opposing ideas, an example for a young child would be eating their vegetables. They don’t really hate eating them, they fear what they would taste like because they’re something new and taboo to a child. We fear the unknown and label that with hatred. Humans are quick to judge and fear the unknown. We may not admit it but we are plagued with xenophic tendencies. Hate creates a chain reaction that many people are not aware of but fall into quite often, with the end result always being fear. The reason for this chain reaction is lack of understanding and immediately gets the feeling of fear which leads to hate. The first reaction of hate is the act of attacking. People will attack a person or an idea through words or actions, â€Å"No one attacks without intent to hurt†. An example of a hate related attack could be directed towards homosexuals. They are given cruel names such as faggot or flamer, which is the attack. The chain reaction is then followed by guilt, guilt because you realize your attack was not logical, For example what if an adult made fun of homosexuals repeatedly only to find out that their own child is actually gay. They would then feel the heavy pressure of guilt. Fear is the end result in this chain reaction. People fear punishment and they fear that because they know their thoughts/actions were not right, for example a parent may fear the punishment of losing their child because of their attacks on something sensitive for the child. The end result of hate is always fear. When someone makes a choice to â€Å"hate† something, someone or an idea they’re really making a choice for fear. Hate makes people feel empowered, when they have the idea that they hate something instilled in their head it makes them feel higher than the object or idea they â€Å"hate† giving them the feeling of being a hierarchy. With the feeling of hate a person also feels stronger. In today’s society it is so much easier for one to say they hate a certain thing and leave it at that rather than have to explain themselves for the reasoning behind the hatred, people hide fear with hatred. As Lester B Pearson once said â€Å"Misunderstanding arising from ignorance breeds fear, and fear remains the greatest enemy of peace†. How to cite Hate vs Fear, Essay examples

History and Politics of the Theories of Development

Question: Discuss about the History and Politics of the Theories of Development. Answer: Introduction In the advancing and developing the world after the WW2 and the 1945 era, the need to modernize the world by the development of modernization theory was a significant advancement in the history of developmentstudies. However, the suggestions put forward by the modernization theories in terms of development and modernity gave rise to controversy immediately after it developed into atheory. This essay would attempt to highlight the meaning of modernization and use Rostows modernization theory of development to analyze the conceptfrom different perspectives and in different stages. While analyzing the theory, the context of the theorys development would also be discussed, while focusing on the perspectives shared by different developmental studies scholars (Rostow, Baker Baker, 2016). The modernization theory had two major objectives attempted to find a cause behind the failure of poor countries in developing and offer a non-communist answer to poverty for developing the world with the help of cultural and economic changes. In this Rostows modernization theory would be critically analyzed in the context of modernization. Rostow opined that any initial insertion of aid from the Western part of the globe, in the form of education, training, and economic investment and so on, would be sufficient for any society to dive towards economic development, after overcoming any barriers. He has suggested the development be viewed in the form of an evolutionary process for countries which progress up the five rung ladder (Kindleberger, Di Tella Di Tella, 2016). Analysis Rostows modernization theory or Rostow's Stages of Growth, also known as the Rostovian take-off modelis a major historical model of economic growth. WW Rostow is the developer of this model, which hypothesizes that economic modernization, happens in five primary stages, which have a varying length (Abdu, 2013). The traditional society largely consists of the population who are devoted to agriculture. The standard of technology in this kind of society is almost restrained and reflects somewhat the pre-Newtonian era.Some forms of society representing this traditional form include the Middle East, theChinese dynasties, the Mediterranean and the ancient civilizations of Europe. The precondition for takeoff stage exists in nations which have a more stable political condition. In this kind of society, there is a larger exploitation of science and technology, additional investments in transportation and communication technologies.There have been appearances of modern manufacturing processes (Elliott, 2012). In the takeoff stage, agriculture is more commercialized with the appearance of new industries. The available natural resources started to get exploited, with the rise and steady growth experienced in savings and investments. The drive to maturity stage in an economy comes after a long phase of development, approximately 40 years, in which around 10-20% of national income is invested, and the resultant output continuously outstrips the population growth. There is a huge change away from heavy engineering processes in the direction of more compound procedures. The economy has the option to choose anything for producing anything it may want, even if the required natural resources are not available. Although 40-60 years is the standard assumption, Rostow opines that the length of the time might vary. The age of high mass consumption sees a large amount of the population moving ahead of meeting their fundamental requirements. Major economy sectors are into the production of durable goods. For instance, the production and launch of Fords Model T indicated the launch of this process in the USA. This age also saw an allocation of the increased resources in social welfare and security (Rostow, 2014). According to the view put forward by Raslow, the advanced countries have all surpassed the takeoff stage and have been able to achieve sustainable growth. The economies in the developing level were either in the preconditioning stage or the traditional stage. Everything that these societieshad to carry out to reach the takeoff stage was to follow a specific set of developmental rules (Musson, 2014). The takeoff stage has been defined by Raslow as a stage in which the degree of productive economic activity attains a critical level and then presents with changes that lead to a large and advanced structural modification of the economy and the society as a whole.The takeoff stage can only be arrived atif three criterions are met with. At first, the nation has to raise its investment rates with the amount of investment being nothing less than 10 percent of the national income.This criterion can be met with either by means of investment from the nation's savings or by foreign investment or support. Secondly, the country needs to develop one or more sizeable manufacturingsectors that have a high growth rate. Finally, there is a need for the existence or creation of a social, political and institutional framework for the promotion of theextension of the new modern sector (Li Hung, 2013). Within this theory of Raslows, the growth of the economy was calculated by an increasing per capita income.In opposition to the structuralists, Rostow was unconcernedabout the productions being evenly distributed amongst all the economic sectors. Therefore, again contrasting the structuralists, Rostow associatedeconomicdevelopment witheconomicgrowth. To encourage the growth, the nation had to boost the investments and savings. Due to the low savings rate in developing nations, it was the responsibility of the government within this theory of creating a group of people who had the tendencyto save. It was also the governments responsibilitytomake sure that those people who saved more received a larger share of the national income. Otherwise, the scenario would be like the national income being consumed entirely instead of being invested (Zamberi Ahmad Xavier, 2012). Due to the comfortability of this model, it was so popular and widely accepted. It was able to justify the huge transfers of technology and capital from the industrialized countries as well as the developing countries. Simultaneously it was able to provide with a rationale for the large concentrations of wealth that was existent in the developing countries (Lewis, 2013). Rostow's stages of budgetary improvement are the most extensively streamed and especially commented bit of commercial written work starting late. It is against Marx's stages of feudalism, bourgeoisie, private endeavour, socialism and communism. In any case, there are inquiries by money related experts in the validity of the division of financial history into five stages of improvement as presented by Rostow. To keep up that every economy takes after a comparable course of headway with a run of the mill past and a near future is to over-schematize the brain boggling qualities of change and to give the gathering of stages a broad proclamation that is stunning (Ranieri Almeida Ramos, 2013). Rostow prescribes capital is required for a country to move from its traditional society (stage 1) to the further stages of change. In many making countries inside Asia and Africa, there have been generous mixtures of cash yet a huge piece of the people are still in the traditional society stage. Countries, for instance, Brazil and Mexico have continued forward to the preconditions for take-off (stage 2) monetarily, yet in doing in that capacity have obtained huge national commitments (Cohen, 2013). Rostow propels that there is a short time span cross between takeoff (stage 2) and maturity (stage 3) when a country winds up evidently self-keeping up. Essentially time scopes of improvement is a significantly more befuddled picture, as a result of the way that making and as of late made countries pick up from fiscally settled countries. Inside this stage the country is self-overseeing, budgetary advancement is spreading, and with its transport, development systems and urbanization make. War and money related approvals can drive the model to a stop or even backward in unusual conditions. This would be suitable to the current political condition in Iraq (Brookfield, 2012). The most impeding presumption that Rostow is rebuked for is endeavoring to fit budgetary advancement into a linear structure. This direction is appropriate in that various countries to make false starts accomplish a level of the move and after that reposition, or simply like the scenario in contemporary Russia, slip again from high mass consumption (or about) to a country encountering critical change. Of course, Rostow's examination seems to underscore accomplishment since it is endeavoring to elucidate accomplishment. To Rostow, if a country can be controlled, uncorrupt monetary authority in it, can incorporate up particular norms with its society and province and can recognize parts where it has some ideal position, it can go into the move and at last accomplish advancement. Rostow would show a malfunction in one of these circumstances as an explanation behind non-linearity (Van Duijn, 2013). Another issue that Rostow's work has is that it believes free countries: countries with an immeasurable people (Japan), with customary resources open at essentially the crucial time in its history (Coal in Northern European countries), or with a broad land mass (Argentina). He has little to state and as a general rule offers little look for after little countries, for instance, Rwanda, which doesn't have such great conditions (Wilson, 2013). Neo-liberal economic theory to Rostow, and various others, offers might want to a remarkable piece of the world that economic ripeness is coming and the season of high mass consumption is close. Regardless, that leaves a sort of 'grim meathook future' for the exemptions, which don't have the advantages, political will, or outside sponsorship to wind up evidently forceful. The task of traditional social requests as pre-Newtonian ignores the dualism of many present-day LDCs (Less Developed Countries). A remarkable piece of the far reaching amassing , home, and mining zones of India, Indonesia, Nigeria and Pakistan are present day techniques and strategies and can't be seen as traditional in Rostow's sense (Hite Chorev, 2014). A great deal of Rostow's suggestion about conditions for takeoff is revoked by observational data. Augments in hypothesis rates and improvement don't occur in the 20-30 year navigate Rostow relegates for takeoff. Improvement in wander rates and exclusive national thing in Great Britain, Germany, Sweden, and Japan show a direct and for the most part persevering expanding speed rather an unforeseen takeoff (Lin, 2012). Rostow's premise that economic modernization proposes a change from a youthful economy to one like those in North America and Western Europe today represents another issue. Rostow dissects LDCs at the opportunity to the improvement of nation stages in the West. The expect that the change of youthful countries will parallel earlier stages of today's impelled countries, in any case, he overlooks the relationship of contemporary juvenile countries with made countries and what's more each LDC's uncommonly particular history. Finally, we may reason that rather than being one way to deal with economic change, there are various. In any case, all around to progression, there are basic qualities, and Rostow has viably recognized some of them (Acs, Szerb Autio, 2016). In any case, the way that Rostow's model has its reporters, which is their conviction framework (their inclination), the less made economies seems to set up a fight against insightful expansionism by the gathered made economies who endeavor to measure them by their theories. Rostow's change exhibit didn't just show the dynamic course from primitive to the inventive progress of nations moreover tried, however, offhand by. Rostow's model still uncovers knowledge into a compelling approach to economic headway for a couple of countries. His model is up 'til now a champion among the most, for the most part, referred to progression theories, and is a primary instance of the union of geography, economics, and administrative issues (Popkova, 2014). Conclusion The recorded scenery of present day common requests is the view that front line countries had passed the stage of take-off into self-overseeing advancement. While the undeveloped countries are so far experiencing traditional society or the pre-condition to take-off. Take note of that takes off is preferably suited for the industrialization of youthful countries. For a country that is making, change of no less than one driving parts is valuable amid the time spent industrialization. In spite of the way that, the original fragments can be in cultivating or in the formation of fundamental things for exports. Having seen, that developing economies are depicted by the predominance of cultivating and basic era. These under making countries are scarcest foreseen that would experience those stages of monetary improvement remembering the ultimate objective to finish its goals. With everything taken into account, youthful countries must take in a lesson from the money related history of bleeding edge nations. They should take after the fundamentals of headway to take-off and after that to self-supporting budgetary improvement. In this note, the youthful countries should actuate family unit, and outside assets to create sufficient dare to revive budgetary improvement this in no little way will lead the country in finishing supportable progression in light of the way that totaled advancement point of fact prompts change. References Abdu, M. (2013). Foreign direct investment and economic growth in Nigeria.International Journal of Arts Sciences,6(1), 63. Acs, Z. J., Szerb, L., Autio, E. (2016). The global entrepreneurship and development index. InGlobal Entrepreneurship and Development Index 2015(pp. 11-31). Springer International Publishing. Brookfield, H. (2012).Interdependent development. Routledge. Cohen, M. J. (2013). Collective dissonance and the transition to post-consumerism.Futures,52, 42-51. Elliott, J. (2012).An introduction to sustainable development. Routledge. Hite, A. B., Chorev, N. (2014).The globalization and development reader: Perspectives on development and global change. John Wiley Sons. Kindleberger, C. P., Di Tella, G., di Tella, G. (2016).Economics in the Long View: essays in honour of WW Rostow. Springer. Lewis, W. A. (2013).Theory of economic growth(Vol. 7). Routledge. Li, R. Y. M., Hung, R. (2013). Rostows Stages of Growth Model,'Urban Bias' and Sustainable Development in India. Lin, J. Y. (2012). New structural economics: a framework for rethinking development. Musson, A. E. (2014).Science, technology and economic growth in the eighteenth century. Routledge. Popkova, E. G. (2014). New prospects of economic growth in context of underdevelopment whirlpools phenomena.Applied Econometrics and International Development,14(1), 5-20. Ranieri, R., Almeida Ramos, R. (2013).Inclusive growth: Building up a concept(No. 104). Working Paper, International Policy Centre for Inclusive Growth. Rostow, W. W. (2014).How it All Began (Routledge Revivals): Origins of the Modern Economy. Routledge. Rostow, W., Baker Jr, R., Baker Jr, R. G. (Eds.). (2016).The economics of take-off into sustained growth. Springer. Van Duijn, J. J. (2013).The long wave in economic life. Routledge. Wilson, R. (2013).Economic development in the Middle East. Routledge. Zamberi Ahmad, S., Xavier, S. R. (2012). Entrepreneurial environments and growth: evidence from Malaysia GEM data.Journal of Chinese Entrepreneurship,4(1), 50-69.