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 ( and ) 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.  P.Deligne, Equation differentiel dans le champ complex, Springer Lecture Note in Mathematics, 163.  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. calculus1.defint( intpath1.segment( complex1.complex_cartesian(0,0),complex1.complex_cartesian(1,0)), OMBIND(fns1.lambda,OMBVAR(z), arith1.power(z,p ~arith1.minus~ 1) ~arith1.times~ arith1.power(1 ~arith1.minus~ z, q ~arith1.minus~ 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 ~arith1.times~ nums1.pi ~arith1.times~ complex1.complex_cartesian(0,1)) ~relation1.eq~ calculus1.defint( intpath1.circle( complex1.complex_cartesian(0,0),1), OMBIND(fns1.lambda,OMBVAR(z), arith1.power(z,arith1.unary_minus(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. ( (c1 ~relation1.eq~ transc1.exp(2 ~arith1.times~ nums1.pi ~arith1.times~ complex1.complex_cartesian(0,1) ~arith1.times~ p)) ~logic1.and~ (c2 ~relation1.eq~ transc1.exp(2 ~arith1.times~ nums1.pi ~arith1.times~ complex1.complex_cartesian(0,1) ~arith1.times~ q)) ~logic1.and~ (e ~relation1.eq~ (1 ~arith1.divide~ 10)) ) ~logic1.implies~ calculus1.defint( (intpath1.circle_with_starting_point( complex1.complex_cartesian(0,0),e,complex1.complex_cartesian(e,0)) ~arith1.divide~ (c1 ~arith1.minus~ 1)) ~arith1.plus~ intpath1.segment(complex1.complex_cartesian(e,0)) ~arith1.minus~ (intpath1.circle_with_starting_point( complex1.complex_cartesian(1,0),e, complex1.complex_cartesian(1 ~arith1.minus~ e,0)) ~arith1.divide~ (c2 ~arith1.minus~ 1)), OMBIND(fns1.lambda,OMBVAR(z), arith1.power(z,p ~arith1.minus~ 1) ~arith1.times~ arith1.power(1 ~arith1.minus~ z, q ~arith1.minus~ 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. calculus1.defint( intpath1.path_in_sector(intpath1.infty,0,0) ~arith1.plus~ intpath1.segment( intpath1.path_in_sector(intpath1.infty,0,0), intpath1.path_in_sector(intpath1.infty, arith1.unary_minus(2 ~arith1.times~ nums1.pi ~arith1.divide~ 3), arith1.unary_minus(2 ~arith1.times~ nums1.pi ~arith1.divide~ 3))) ~arith1.plus~ intpath1.path_in_sector(intpath1.infty, arith1.unary_minus(2 ~arith1.times~ nums1.pi ~arith1.divide~ 3), arith1.unary_minus(2 ~arith1.times~ nums1.pi ~arith1.divide~ 3)), OMBIND(fns1.lambda,OMBVAR(t), transc1.exp(t ~arith1.minus~ (x ~arith1.times~ arith1.power(t,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).