The Beta Function

The beta function is defined by B(x,y) = I0,1(t^(x-1)(1-t)^(y-1) dt, where the real parts of x and y are greater than 0.
This function can also be expressed in terms of the gamma function, through the formula B(x,y) = G(x)G(y)/G(x+y).

Other important relationships include:
B(x,y) = B(y,x)
B(x,y) = 2I0,pi/2((sin(t))^(2x-1)(cos(t))^(2y-1) dt
B(x,y) = I0,infinity(t^(x-1)/(1+t)^(x+y)
B(x,y)B(x+y,1-y) = pi/(x sin(pi y)) 

Also, the binomial coefficient can be defined in terms of the beta function through the formula (n,k) = 1/((n+1)B(n-k+1,k+1)), where (n,k) is the binomial coefficient of n and k.

The Gamma Function

The gamma function of z.  In my discussion I will use G(z) to represent it.

The absolute value of the gamma function on the complex plane.

The gamma function, G(x) can be seen as a line connecting points given by the equation y = (x-1)! for integer values of x, where ! is the factorial function.  Thus, for integers, G(x) = (x-1)!.  Other important properties of the gamma function include:
G(1-x)G(x) =  pi/sin(pi x)
G(x)G(x+.5) = 2^(1-2x) pi^.5 G(2x)
G(x)G(x+1/m)G(x+2/m)..G(x+(m-1)/m) = (2pi)^((m-1)/2) m^(.5-mx) G(mx)

The derivative of the gamma function can be shown in terms of the polygamma function, P0(x), with the relationship G`(x) = G(x)P0(x).  This can be generalized into the form d^n/dx^n(G(x)) = I0,infinity(t^(x-1)e^(-t)(ln t)^n dt, where I0,infinity is the definite integral from 0 to infinity.

Complex Integrals

Sometimes, integrals cannot be solved in terms of the elementary integral shortcuts.  When this occurs, techniques such as substitution, integration by parts, or partial fractions.

As an example, I will show how to integral f(x) = (4-x^2)^1.5.
First, x must be substituted for u, with the equations x = 2sin(u) and dx = 2cos(u) du.
Then, I(f(x)) = I((2cos(u))^3(2cos(u)) du) = 16 I(cos(u)^4 du)
This can be simplified using the identity cos(u)^2 = (1+cos(2u))/2
Using the identity, I(f(x)) = 16 I((1+cos(2u))(1+cos(2u))/4) du)
Distributing, I(f(x)) = 4 I(1 + 2cos(2u) + cos(2u)^2 du)
Using the identity cos(u)^2 = (1+cos(2u))/2 again, I(f(x)) = 4 I(1 + 2cos(2u) + (1+cos(4u))/2 du)
Distributing, I(f(x)) = 2 I(3 + 4cos(2u) + cos(4u) du)
This can be simplified using the equations a = 2u, da = 2du, b = 4u, db = 4du
Then, I(f(x)) = 6 I(du) + 4 I(cos(a) da) + 1/2 I(cos(b) db)
Evaluating, I(f(x)) = 4sin(a) + 1/2sin(b) + 6u
Substituting for u, I(f(x)) = 4sin(2u) + 1/2sin(4u) + 6u
Using the identity sin(2u) = 2sin(u)cos(u), I(f(x)) = 8sin(u)cos(u) + 2sin(2u)cos(2u) + 6u
Using the identity cos(2u) = 1 - 2sin(u)^2, I(f(x)) = 8sin(u)cos(u) + 4sin(u)cos(u)(1-2sin(u)^2) + 6u
Distributing, I(f(x)) = 8sin(u)cos(u) + 4sin(u)cos(u) - 8sin(u)^3cos(u) + 6u
Using the identity cos(u) = (1-sin(u))^.5, I(f(x)) = 8sin(u)(1-sin(u)^2)^.5 + 4sin(u)(1-sin(u)^2)^.5 - 8sin(u)^3(1-sin(u)^2)^.5 + 6u
Substituting for x, I(f(x)) = 4x(1-1/4x^2)^.5 + 2x(1-1/4x^2)^.5 - x^3(1-1/4x^2)^.5 + 6arcsin(1/2x)
Factoring, I(f(x)) = (6x-x^3)(1-1/4x^2)^.5 + 6arcsin(1/2x)
Finally, I(f(x)) =1/4x(6-x^2)(4-x^2)^.5 + 6arcsin(1/2x)

Integrals and Integral Shortcuts

An example of an integral.  I will be using "I" instead of the integral sign at the beginning.

If the integral of a function f(x) is f'(x), the integral of the function is f(x).  Thus, the integral is the "antiderivative" of f'(x).

The shortcuts for integrals are usually more helpful than those for derivatives, since integrals are more complex to calculate by hand.  Some shortcuts include:

I(u^n du) = u^(n+1)/(n+1) when n != -1
I(du/u) = ln(u)
I(sin(u) du) = -cos(u)
I(cos(u) du) = sin(u)
I(tan(u) du) = ln|sec(u)|
I(cot(u) du) = ln|sin(u)|
I(sec(u) du) = ln|sec(u) + tan(u)|
I(csc(u) du) = ln|csc(u) - cot(u)|
I(sec(u)^2 du) = tan(u)
I(csc(u)^2 du) = -cot(u)
I(sec(u)tan(u) du) = sec(u)
I(csc(u)cot(u) du) = -csc(u)
I(a^u du) = a^u/ln(a)
I(e^u du) = e^u
I(sinh(u) du) = cosh(u)
I(cosh(u) du) = sinh(u)
I(tanh(u) du) = ln(cosh(u))
I(coth(u) du) = ln|sinh(u)|
I(sech(u) du) = arctan(sinh(u))
I(csch(u) du) = -arccoth(cosh(u))
I(sech(u)^2 du) = tanh(u)
I(csch(u)^2 du) = -coth(u)
I(sech(u)tanh(u) du) = -csch(u)
I(du/(a^2-u^2)^.5 = arcsin(u/a)
I(du/(u^2+-a^2)^.5) = ln|u + (u^2 +- a^2)^.5|
I(du/(u^2+a^2)) = 1/a arctan(u/a)
I(du/(u^2-a^2)) = 1/(2a) ln|(u-a)/(u+a)|
I(du/(u(a^2+-u^2)^.5)) = 1/a(ln|u/(a+(a^2+-u^2)^.5)|
I(du/(u(u^2-a^2)) = 1/a arccos(a/u)
I((u^2+-a^2)^.5 du) = u/2(u^2+-a^2)^.5 +- a^2/2 ln|u + (u^2+-a^2)^.5|
I((a^2-u^2)^.5 du) = u/2(a^2-u^2)^.5 + a^2/2 arcsin(u/a)
I(e^(au) sin(bu) du) = e^(au)(a sin(bu) - b cos(bu))/(a^2+b^2)

Partial Derivatives

Partial differentiation is the same as regular differentiation, but it keeps all variables in a multivariable function the same except for one variable.  The partial derivative d/dx(f(a,b,c,...x,y,z)) can be defined through the limit definition: d/dx(f(a,b,c,...x,y,z)) = lim(h->0) (f(a,b,c...,x + h,y,z) - f(a,b,c,...,x,y,z))/h.  For example, the derivative of f(x, y)=xy is lim(h->0) (f(x + h,y) - f(x,y))/h) = lim(h->0) (((x + h)y - xy)/h) = lim(h->0) ((xy + hy - xy)/h) = lim(h->0) (hy/h) = y.

Shortcuts for Differentiation

While the limit definition of the derivative can allow differentiation, it is often tedious and the shortcuts for differentiation can save time.  Some shortcuts include:

d/dx(C)=0
d/dx(u^n) = nu^(n-1) du/dx
d/dx(sin(u)) = cos(u) du/dx
d/dx(cos(u)) = -sin(u) du/dx
d/dx(tan(u)) = sec(u)^2 du/dx
d/dx(cot(u)) = -csc(u)^2 du/dx
d/dx(sec(u)) = sec(u)tan(u) du/dx
d/dx(csc(u)) = -csc(u)cot(u) du/dx
d/dx(logau) = (logeu)/u du/dx
d/dx(ln(u)) = 1/u du/dx
d/dx(a^u) = a^u ln(a) du/dx
d/dx(e^u) = e^u du/dx
d/dx(arcsin(u)) = 1/(1-u^2)^.5 du/dx
d/dx(arccos(u)) = -1/(1-u^2)^.5 du/dx
d/dx(arctan(u)) = 1/(1+u^2) du/dx
d/dx(arccot(u)) = -1/(1+u^2) du/dx
d/dx(arcsec(u)) = +-1/(u(u^2-1)^.5) du/dx (+ if u > 1 or - if u < -1)
d/dx(arccsc(u)) = +-1/(u(u^2-1)^.5) du/dx (- if u > 1 or + if u < -1)
d/dx(sinh(u)) = cosh(u) du/dx
d/dx(cosh(u)) = sinh(u) du/dx
d/dx(tanh(u)) = sech(u)^2 du/dx
d/dx(coth(u)) = -csch(u)^2 du/dx
d/dx(sech(u)) = -sech(u)tanh(u) du/dx
d/dx(csch(u)) = -csch(u)coth(u) du/dx
d/dx(arcsinh(u)) = 1/(1+u^2)^.5 du/dx
d/dx(arccosh(u)) = 1/(u^2-1)^.5 du/dx
d/dx(arctanh(u)) = 1/(1-u^2) du/dx
d/dx(arccoth(u)) = 1/(1-u^2) du/dx
d/dx(arcsech(u)) = 1/(u(1-u^2)^.5) du/dx
d/dx(arccsch(u)) = -1/(u(u^2+1)) du/dx

Limit Definition of the Derivative

The slope of the line tangent to a curve can be represented by the equation of the derivative.

The derivative is one of the fundamentals of calculus; it shows how a function changes as its input changes. It can be thought of as the slope of the tangent line on a graph on the xy plane, or as the velocity of an object if the position of the object has a relationship to the time, t. The derivative (f'(x)) can be defined as the slope of a secant as the difference in the x values of the endpoints of the secant approaches 0 (i.e. f'(x) = lim(h->0) (f(x + h) - f(x))/h).  For example, the derivative of f(x)=x is lim(h->0) (f(x + h) - f(x))/h) = lim(h->0) ((x + h - x)/h) = lim(h->0) (h/h) = 1.