Website owner: James Miller
Special functions and their transforms. Gamma function. Bessel functions. Error function. Complementary Error function. Exponential integral. Sine and cosine integrals. Dirac delta function. Unit Step Function. Null function.
Def. Gamma function. For positive values of x, the Gamma function is defined by the improper integral
This is Euler’s second integral. It converges for all positive, real values of x. An important property of the gamma function is the relationship
1) Γ(x + 1) = xΓ(x) .
The gamma function can be defined for negative values of x using 1) i.e.
For more information see Gamma function.
Bessel functions of the first kind. Let ν be a real number. The function
is known as the Bessel function of the first kind of order ν. The formula is valid providing ν -1, -2, -3, .... . Γ(ν) is the gamma function.
For more information see Bessel functions.
Error function. The error function is defined as
Complementary Error function. The complementary error function is defined as
Exponential integral. The exponential integral is defined as
Sine integral. The sine integral is defined as
Cosine integral. The cosine integral is defined as
Dirac delta function. The Dirac delta function δ(t), also called the impulse function, is a function which is zero for every value of t except at t = 0, where it is infinite in such a way that
Consider the function Fε(t)
consisting of a pulse of width ε and height 1/ε shown in Fig. 1. The Dirac delta function can be viewed as the limit of Fε(t) as the width ε of the pulse approaches zero. Two of its properties are
The Dirac delta function is not, strictly speaking, a function but it is treated as a function, manipulated as a function.
Unit Step Function. The unit step function is defined as
Syn. Heaviside’s unit function
Fig. 1a shows a graph of y = u(t). Fig. 1b shows a graph of y = u(t - a)
Remark. The function g(t) = f(t)u(t - a) represents the function f(t) as cut off to the left of t = a.. See Fig. 3.
Null function. If N(t) is a function of t such that for all t > 0
we call N(t) a null function.
Example. The function
is a null function.
In general, a function that is zero at all but a countable set of points is a null function.
Transforms of special functions
Murray R. Spiegel. Laplace Transforms. (Schaum)
Murray R. Spiegel. Mathematical Handbook. (Schaum)
C. R. Wylie, Jr. Advanced Engineering Mathematics.
The Way of Truth and Life
God's message to the world
Jesus Christ and His Teachings
Words of Wisdom
Way of enlightenment, wisdom, and understanding
Way of true Christianity
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
Who will go to heaven?
The superior person
On faith and works
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
The teaching is:
On modern intellectualism
On Self-sufficient Country Living, Homesteading
Principles for Living Life
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
America has lost her way
The really big sins
Theory on the Formation of Character
You are what you eat
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
These things go together
We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
The True Christian
What is true Christianity?
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Walking a solitary road
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
On responding to wrongs
Real Christian Faith
The Natural Way -- The Unnatural Way
Wisdom, Reason and Virtue are closely related
Knowledge is one thing, wisdom is another
My views on Christianity in America
The most important thing in life is understanding
Sizing up people
We are all examples --- for good or for bad
Television --- spiritual poison
The Prime Mover that decides "What We Are"
Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life
We are our habits
What creates moral character?