intpath1
http://www.math.kobe-u.ac.jp/OCD/intpath1.ocd
2003-07-08
experimental
2004-07-08, 2004-12-12
1
1
arith1
calculus1
complex1
fns1
list1
logic1
nums1
relation1
set1
transc1
This CD defines symbols to express pathes for integration
for complex integral in one variable.
Our pathes lie in the one dimensional complex projective space =
the Riemann sphere.
These pathes are sufficiently rich to express most pathes of integrations
in the complex special function theory in one variables.
The theory of twisted cycles ([1] and [2]) answers to the question
of telling the exact space of cycles (pathes) defined by symbols in this
CD.
Our symbols are enough to express integral pathes
when integrand functions are solutions of ordinary differential equations
with polynomial coefficients.
The details will be discussed in a separate paper.
[1] P.Deligne, Equation differentiel dans le champ complex,
Springer Lecture Note in Mathematics, 163.
[2] K.Matsumoto, H.Majima, N.Takayama,
Quadratic relations for confluent hypergeometric functions.
Tohoku Mathematical Journal 52 (2000), 489--514.
infty
The infty on the Riemann sphere. When the coordinate of the complex plane
is z, we call t=1/z the standard coordinate around the infinity
of the Riemann sphere.
segment
The symbol segment(a,b) is the segment from the point a to the point b
in the complex plane.
If the arguments are sectors given by path_in_sector, it means the segment
from a point in the circular border of the sector to a point
in the circular border of the another sector.
The beta function B(p,q) [Re(p)>0, Re(q)>0] is defined by
the following integral.
0
0
1
0
1
1
1
circle
The symbol circle(c,r) is the circle in the Riemann sphere of which
center is c and the radius is r.
The direction of the circle is the counter clockwise.
When the center is intpath1.infty, the radius should be given in the
standard coordinate t=1/z at the infinity.
The residue of 1/z is equal to 2 pi sqrt(-1).
2
0
1
0
0
1
1
circle_with_starting_point
The symbol circle_with_starting_point(c,r,z0) is the circle
in the Riemann sphere of which center is c and the radius is r.
The direction of the circle is the counter clockwise and
the staring point is z0.
The integral representation of the beta function B(p,q) by the twisted cycle
is as follows.
2
0
1
2
0
1
1
10
0
0
0
1
0
1
0
1
0
1
1
1
1
path_in_sector
The symbol path_in_sector(c,t1,t2) is an outgoing path
in a sufficiently small sector with the center c and the angles t1 and t2.
The path starts from the point c and it is sufficiently short.
When the center is intpath1.infty, the angle should be given in the coordinate
t=1/z.
The Airy integral is expressed as follows.
0
0
0
0
2
3
2
3
2
3
2
3
3
path_in_sector2
The symbol path_in_sector2(c,t1,t2,z0) is an outgoing path
in the sector with the center c and the angles t1 and t2.
The path is the segment from the point c to the point z0 which
lies in the sector.
closed_path
The symbol closed_path(start_end,points_in, points_out) is a closed
path with the starting point "start_end".
The direction of the path is counter clockwise.
It contains the set of points "points_in" in the inside of the path.
The winding number of the path for each point in the set points_in
is 1.
The path is zero homotope in the space (P^1 - points_in).