erf xdx x erf x x 2 ex2 dx C
x erf x 1 ex2 C
The above equation is tabulated under the symbol “ ierf x” with C 1
(Therefore, ierf 0 = 0)
Another related function is the complementary error function “erfc x”
– We have studied many elementary functions such as polynomials, powers, logarithms, exponentials, trigonometric and hyperbolic functions.
– Four kinds of Bessel functions are useful for expressing the solutions of a particular class of differential equations.
2 ez2
erf x
erf x 2 x ez2 dz
0
z: dummy variable
0x
z
Proof in next slide
1
I R ex2 dx R e y2 dy x and y are two independent Cartesian coordinates
0
0
I 2 R R e(x2 y2 )dxdy 00 in polar coordinates
I 2 R
1 2
er2 rdrd
00
1 R2eR2
2
I 2 1 1 eR2
44
Error between the volume determined by x-y and r-
The volume of has a base area which is less than 1/2R2 and a maximum height of e-R2
– Legendre polynomials are solutions of a group of differential equations. Learn some more now….
The error function
• It occurs in the theory of probability, distribution of residence times, conduction of heat, and diffusion matter:
R , I 2 1
4
erf x 2 x ez2 dz
0
erf 1
More about error function
erf x
2 x ez2 dz
0
Differentiation of the error function: d erf x 2 ex2
dx
Integration of the error function:
(n) t e n1 t dt 0
t n e 1 t
0
(n 1)
t n2etdt
0
(n 1)(n 1)
repeat
(n) (n 1)(n 2)...( 2)(1)(1) (n 1)!
The gamma function is thus a generalized factorial, for positive integer values of n, the gamma function can be replaced by a factorial.
(Fig. 5.3 pp. 147)
More about the gamma function
1
t
1 2
e
t
dt
t x2
1 x1ex2 2xdx 2 ex2 dx
2 0
dt 2xdx 2 0
0
1 erf
2
Evaluate
3
1
2
3 1 2
(n) (n 1)(n 1)
erfc x 1 erf x 2 ez2 dz
x
The gamma function
(n) t e n1 tdt 0
for positive values of n. t is a dummy variable since the value of the definite integral is independent of t. (N.B., if n is zero or a negative integer, the gamma function becomes infinite.)
5 2 1 5 3 1 1 5 3 1 2 2 2 2 2 2 2 2 2
化工應用數學
授課教師： 郭修伯
Lecture 6
Functions and definite integrals Vectors
Chapter 5
Functions and definite integrals
There are many functions arising in engineering which cannot be integrated analytically in terms of elementary functions. The values of many integrals have been tabulated, much numerical work can be avoided if the integral to be evaluated can be altered to a form that is tabulated.
Ref. pp.153
We are going to study some of these special functions…..
Special functions
• Functions
– Determine a functional relationship between two or more variables