Calculations With the Gamma Function

The gamma function is defined by the following complicated looking formula: Γ ( z ) ∠«0∞e - ttz-1dt One question that people have when they first encounter this confusing equation is, â€Å"How do you use this formula to calculate values of the gamma function?†Ã‚  This is an important question as it is difficult to know what this function even means and what all of the symbols stand for. One way to answer this question is by looking at several sample calculations with the gamma function.  Before we do this, there are a few things from calculus that we must know, such as how to integrate a type I improper integral, and that e is a mathematical constant.   Motivation Before doing any calculations, we examine the motivation behind these calculations.  Many times the gamma functions show up behind the scenes.  Several probability density functions  are stated in terms of the gamma function. Examples of these include the gamma distribution and students t-distribution,  The importance of the gamma function cannot be overstated.   Γ ( 1 ) The first example calculation that we will study is finding the value of the gamma function for Γ ( 1 ). This is found by setting z 1 in the above formula: ∠«0∞e - tdt We calculate the above integral in two steps: The indefinite integral ∠«e - tdt -e - t CThis is an improper integral, so we have ∠«0∞e - tdt limb → ∞ -e - b e 0 1 Γ ( 2 ) The next example calculation that we will consider is similar to the last example, but we increase the value of z by 1.  We now calculate the value of the gamma function for Γ ( 2 ) by setting z 2 in the above formula. The steps are the same as above: Γ ( 2 ) ∠«0∞e - tt dt The indefinite integral ∠«te - tdt- te - t -e - t C.  Although we have only increased the value of z by 1, it takes more work to calculate this integral.  In order to find this integral, we must use a technique from calculus known as integration by parts. We now use the limits of integration just as above and need to calculate: limb → ∞ - be - b -e - b -0e 0 e 0. A result from calculus known as L’Hospital’s rule allows us to calculate the limit limb → ∞ - be - b 0. This means that the value of our integral above is 1. Γ (z 1 ) zΓ (z ) Another feature of the gamma function and one which connects it to the factorial is the formula Γ (z 1 ) zΓ (z ) for z any complex number with a positive real part. The reason why this is true is a direct result of the formula for the gamma function. By using integration by parts we can establish this property of the gamma function.

Christ in Discipleship - 1731 Words

Christ In Discipleship DSMN 500: Discipleship Ministries March 27, 2014 The definition of a disciple is â€Å"One who accepts and assists in spreading the doctrines of another, as one of the twelve in the inner circle of Christ s followers according to the Gospel accounts.† Discipleship means nothing without Christ being the central focus. It is crucial that Christ be the central focal point for all Christian discipleship. There are many ways to disciple someone, but there is only one means of discipleship. Hull states in his book, The Disciple Making Pastor, â€Å"Disciple making takes more faith than any other task of the church.† Not only does it take†¦show more content†¦For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation.† 2 Corinthians 5:17 (KJV) â€Å"Therefore if any man be in Christ, he is a new creature: old things are passed away; behold, all things are become new.† There are four steps that are involved in the stage of declaration. Curious, convi cted, convinced, and converted are the steps that have to take place according to Dempsey and Earley. When God begins to deal with a person’s heart, Jesus will be a curiosity to them. This is where they might visit a church or Bible study group. Then as they become more curious, God’s convicting power starts to stir their hearts and convict him or her of the sin in their life. It is at this point that one of two things will happen. The person will either ignore God’s call and the conviction of their hearts, or they will be convinced God is real and alive and that He wants to forgive them. Once the sinner is convinced, it should not be much longer before they take the final step of conversion. Conversion is where the person surrenders to the drawing and calling of the Holy Spirit and asks for forgiveness of their sins. Once repentance and conversion take place, the first stage of discipleship is completed. The second stage is when development and spiritual growth take place. 2 Peter 3:18 (KJV) â€Å"But grow in grace, and in the knowledge of our Lord and Saviour JesusShow MoreRelatedChrist in Discipleship1938 Words   |  8 PagesLIBERTY UNIVERSITY SCHOOL OF DIVINITY Writing Assignment 1 – Christ in Discipleship Submitted to Dr. Gary Waller in partial fulfillment of the requirements for the completion of DSMN 500 – B03 Discipleship Ministries by David J. Miller May 29, 2016 Centrality of Christ in Christian discipleship Christian discipleship is to follow Jesus, to be his disciple, doesn’t mean community involvement and the veneer of tolerance. It means, mainly, first and central, to worship himRead MoreJesus Christ As The Center Of Discipleship1496 Words   |  6 PagesWhen someone think about Jesus Christ being the center of discipleship there are two things that comes to mind; change and commitment within the disciple. One day Jesus Christ was walking by the sea of Galilee when He saw Simon and Andrew fishing; he said to them, â€Å"Follow me and I will make you fishers of men† Matthew 4:19 . The next verse show them immediately dropping their nets and followed Jesus Christ. 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What is Man in Civilized Society Free Essays

string(39) " they were totally imperfect and rude\." Jean Jacques Rousseau was a French philosopher whose disposition on human nature took him into the vantage of man’s birth and his existence on this earth. Rousseau’s Discourse On Inequality is a development of the man from freedom to his virtual and self created slavery. In a quest to attain the materialistic possessions, he had lost his own freedom, which was earlier, the part of his existence. We will write a custom essay sample on What is Man in Civilized Society? or any similar topic only for you Order Now Man in his â€Å"state of nature† slowly developed himself into a state of artificiality. His whole existence depends upon the fulfillment of his unending needs on the basis of which lies his relationship with others in the society and in environment. This dependency makes him slave to the others for whom he matters the most. This aspect of human nature Rousseau discourses on the Origin of Inequality. All men are born equal yet there is inequality among them and how this inequality among people originated, the answer to it is found in our history. When the first man, after getting hold of a piece of ground, confirmed with pride, This is mine, (Rousseau, 431) and found other people believed him; he had already laid foundation of civilized society. It is a society whereby all men are striving to attain maximum fruits provided by nature but it is also a fact that this is not possible without the help of others. Every one is isolated yet all of them are mutually dependent upon each other and subsequently these are reciprocal needs that unite them. It is also known fact that it is almost very difficult to put a man under subjugation unless he is first put into the position where by he is incapable of doing things without others. Before the man had started learning to work in groups, the first thing man mostly cared for was his own matter of existence and another was his own self preservation. He only needed whatever the nature provided him with and he learned it with his own instincts. Hunger and eagerness to know what is happening around him exposed him to different other ways to sustain on this Earth and among them was the way by which he could propagate his species. This act in him was quite insensitive in nature and without passion. When they were satisfied, both the sexes need not know each other and even children forgot their mothers. Primitive man was not totally ignorant about the ways to utilize natural resources to his advantage but soon he got exposed to many difficulties and the time came when he had to learn various ways to counter these difficulties like fruit laden tall trees making it impossible for him to collect them; competition with other groups who were more eager to collect the fruits from these trees; and the need to counter furious looking and dangerous tribes man. He adopted his own routine exercises as he had to be very swift, and very vigrous in nature with strong physique. He started the use of stones and sticks to create weapons and with his new profound discovery, man embarked his journey to conquer all the obstacles that came in his way and in many situations indemnified himself as a consequence of his defeat against much stronger than himself. As human race grew in large number, man’s needs also grew and subsequently there were number of changes in his way of living. These changes could be due to long winters and extremely hot summers which destroyed the fruits. These hardships might have induced the man to discover something new and this led to formation of an industry. At the seashore, man invented hook line. He also became fish eater and consumed a large number of fish. Then in the forests, he invented bows and arrows and became warrior. Lightenning or volcano must have induced in him the idea of generating electricity. First he learned to preserve whatever knowledge or expertise he considered useful to him, like learning ways to preserve the elements of nature, later the way to reproduce these elements and the method to cook the flesh of animals on fire he was initially eating raw and slowly he was able to overcome all the obstacles posed by the nature to maintain his survival. As more civilizations made their appearance felt across the globe, man’s dangers increased and he began to show more intensity towards his protection and care. All his characteristics like strength, weakness, slow speed, fearfulness and boldness changed according to the needs and increased his intensity to take the precautions necessary for his security. These traits in him also increased his power of reasoning according to the nature and the rules of his own area. He began to judge others in the same way he would judge himself and understood the way they thought and acted in conformity to his own. The experiences also taught him the love for others and the complexities in various relationships. He began to understand how their mutual interests would satiate in his dependence on the other men of his own area, and he also began to realize how their conflicting interests would make him suspicious of others. In the case where he thought his mutual interest lied, he would create temporary association with others, where no member has any restrictions and in later case, every one would seek their own self interest. In this way, men began to understand to work in a mutual cooperation with each other and advantages in fulfilling them and slowly they began to understand to adopt language. For them at that stage, the language of rooks or monkeys was enough to help them communicate with each other. Through cries, gestures and with some sounds, they were able to send their messages across and with the passing of time, they began to articulate the sounds to form their own languages but they were totally imperfect and rude. You read "What is Man in Civilized Society?" in category "Papers" These advancements led them further to develop themselves and now they neither slept under the tree nor in the caves but they invented hard and sharp stones; they would dug inside the earth, cut wood and carved out huts from them. This was the beginning of the first revolution but with this revolution also began the human’s journey towards his very complex and hard relations with the other human beings. Stronger men were able to build their huts first , followed by weaker ones who found themselves safer to imitate them. This was the beginning of the society, when human beings started feeling various emotions that would attach them to each other and from then on, there was a beginning of family relationships, humanity, love and affection. Each family became a small society, and united to each other to attain maximum satisfaction in their personal lives and in their union with each other. The women confined in their homes and men went to the world outside to sustain their lives. This was the beginning of dependency not only among opposite sexes but also among other fellow beings. With this, their needs also began to increase and to meet these needs, they had to adopt and acquire different qualities and skills in relationship to others and also among each other. As said by Rousseau, â€Å"the society in its beginning stages and the relations already established among men required in them qualities different from those they derived from their primitive constitution. †(Rousseau, 435) Now the men were no longer contended with rustic huts, they were not making their own clothes with skins of animals or thorns of fish bones but required labor of others to produce for them luxuries they never dreamt of. This need led to the requirement of labor and dependency of man on each other. Each man relied on each other for fulfillment of his unending needs and desires. Population also began to rise and many civilizations developed which acquired lands and were confined within the precinct of their own boundaries, but this was not a limit because civilizations continued to grow and with each growth, they acquired more and more land. The lust of acquisitions for more land and hope for a revenue from agriculture made it the most costly and valuable commodity. More and more forests began to be transformed into fields needed to be watered and tilled with men’s sweat and with this â€Å"slavery and misery were soon seen to germinate and grow with the crops. (Rousseau, 436). Metallurgy and agriculture and the lust for more possession of these two forms of art developed jealously and pride in the man. The necessity for gaining the knowledge of other art forms arose to give agriculture a boost. While one man was needed to forge the iron, the other was needed to feed them. When workers increased in metallurgy, the workers in the agriculture got reduc ed, as they got involved in extraction of metals; and since some needed food in exchange of iron, barter system came into existence- food in exchange of product and vice versa. The things would have been in the state of the balance if the equal amount of food and minerals had been developed but this was not the case. The people who were stronger in the society were doing majority of work; the skillful of all were turning this opportunity to better their position; and the most ingenious were finding the way to reduce the labor, for e. g. farmer needed iron the most and the blacksmith needed wheat; and if both of them performed equal amount of labor but by putting equal amount of labor, one earned a lot while the other hardly had enough to eat. Thus it is that natural inequality imperceptibly manifests itself together with inequality occasioned by the socialization process. † (Rousseau, 437) It is these occupational differences which enabled men to be more noticeable and more effective, and influenced the fate of others. Man began to develop their own imagination and mental power to decide on several activities. He fruitfully developed several forms of art for his own purpose and the development of these various arts gave him his own rank and position in his own area of working. And he had established himself not just on the basis of the production of goods and services but also on the virtue of his own qualities like beauty, clever mind, strength, skills, and also his other merits and talents. These qualities in him enticed him to prove himself in the society as best in terms how he had made himself and the society to move forward economically, socially and independently, but he too began to show himself more than what he could achieve to his advantage. This trait in him became quite a dangerous proposition as it entails ostentation, deceptiveness, cunningness and all the vices that would make a man slave. Rousseau was quite right when he said that â€Å"On the other hand, although man had previously been free and independent, we find him, so to speak, subject, by virtue of a multitude of fresh needs, to all of nature and particularly to his fellowmen, whose slave in a sense he becomes even in becoming their master; rich, he needs their services; poor he needs their help; and being midway between wealth and poverty does not put him in a position to get along without them†. (Rousseau, 437) What followed was based on the simple fact that man is a social animal and required one person or the other to make him meet and fulfill his needs. But when the most powerful and the weakest try to make the most of each other’s strengths and weaknesses what entails is the unbridled passion for the usurpations of wealth by whatever means and this greed, lust and power make one man dominate the other, still there is no possibility of going on without each other. This is an essence of human civilization and it holds true in 21st century also no matter how advanced scientifically and technologically it is, the man is and would remain slave to his fellowmen. How to cite What is Man in Civilized Society?, Papers