InputMode:=Word

PrecisionDigits:=16

NotationDigits:=16

"Calculate the perimeter of this ellipse."

x^2/a^2+y^2/b^2=1

[a>0,b>0,a>b]

"Calculate one fourth the perimeter and multiply it by four."

[x>=0,y>=0]

;Simp(User)
y=b*SQRT(a^2-x^2)/a

ds^2=dy^2+dx^2

ds=SQRT(dx^2+dy^2)

ds=SQRT(dx^2+dy^2)

;Simp(User)
dy=-b*dx*x/(a*SQRT(a^2-x^2))

;Simp(User)
dy^2=b^2*dx^2*x^2/(a^2*(a^2-x^2))

;Simp(User)
dx^2+dy^2=-b^2*dx^2/(x^2-a^2)-b^2*dx^2/a^2+dx^2

ds=SQRT(-b^2*dx^2/(x^2-a^2)-b^2*dx^2/a^2+dx^2)

;Simp(User)
ds=dx*SQRT((x^2*(a^2-b^2)-a^4)/(x^2-a^2))/a

perimeter=4*INT(SQRT((x^2*(a^2-b^2)-a^4)/(x^2-a^2)),x,0,a)/a

"The perimeter of an ellipse is an elliptic integral."

"Change variables so it will be a elliptic integral in the table."

t=x^2

x=SQRT(t)

dx=dt/(2*SQRT(t))

;Simp(Sub(#21))
t=a^2

perimeter=4*INT(SQRT((t*(a^2-b^2)-a^4)/(t-a^2))/(2*SQRT(t)),t,0,a^2)/a

[a>0,b>0,a>b]

perimeter=2*INT(SQRT(a^4-t*(a^2-b^2))/(SQRT(t)*SQRT(a^2-t)),t,0,a^2)/a

"Before finding the integral in the table, do a numerical approximation."

[a=7.5,b=5]

;Sub(#27)
perimeter=2*INT(SQRT((7.5)^4-t*((7.5)^2-5^2))/(SQRT(t)*SQRT((7.5)^2-t)),t,0,(~
7.5)^2)/(7.5)

;Approx(#30)
perimeter=39.66344269140738

"First set i=1, n=3, and h=3."

IHAT(x):=

AHAT(e):=

"The zeros of this rational expression are:"

[0,a^2,a^4/(a^2-b^2)]

[a>0,b>0,a>b]

ei:=

"The integral is of the type IHAT(e3)."

perimeter=2*(b^2-a^2)*IHAT(e3)/a

"Check this equation."

[a=[0,a^2,a^4],b=[1,-1,-(a^2-b^2)],m=[0,0,1]]

II_AUXH(h,a,b,t):=PRODUCT((a SUB j_+b SUB j_*t)^(-1/2),j_,1,h)

II_AUXN(m,n,a,b,t):=PRODUCT((a SUB j_+b SUB j_*t)^m SUB j_,j_,1,n)

II_AUX(m,h,n,a,b,t):=II_AUXH(h,a,b,t)*II_AUXN(m,n,a,b,t)

AF(m,h,n,a,b,x,y):=II_AUX(m,h,n,a,b,x)-II_AUX(m,h,n,a,b,y)

II(m,h,n,a,b,x,y):=INT(II_AUX(m,h,n,a,b,t_),t_,y,x)

IHAT(e3)=II([0,0,1],3,3,[0,a^2,a^4],[1,-1,-(a^2-b^2)],a^2,0)/(-(a^2-b^2))

;Simp(#48)
IHAT(e3)=INT(SQRT(t_*(b^2-a^2)+a^4)/(SQRT(t_)*SQRT(a^2-t_)),t_,0,a^2)/(b^2-a^~
2)

perimeter=2*(b^2-a^2)*IHAT(e3)/a

;Simp(User)
perimeter=2*INT(SQRT(t_*(b^2-a^2)+a^4)/(SQRT(t_)*SQRT(a^2-t_)),t_,0,a^2)/a

"The equation matches."

[a=7.5,b=5]

;Simp(Sub(#51))
perimeter=2*SQRT(5)*INT(SQRT(405-4*t_)/(SQRT(t_)*SQRT(225-4*t_)),t_,0,225/4)/~
3

;Approx(#54)
perimeter=39.66344269140632

"Look up this integral in the table."

IHAT(ei)=2*AHAT(ej+ek)+rij*rik*IHAT(-ei)

IHAT(e3)=2*AHAT(e1+e2)+r31*r32*IHAT(-e3)

"Calculate AHAT(e1 + e2)."

AHAT(e1+e2)=AF([1,1,0],3,3,[0,a^2,a^4],[1,-1,-(a^2-b^2)],x,y)/(b^2-a^2)

;Simp(#60)
AHAT(e1+e2)=(SQRT(y)*SQRT(a^2-y)*SQRT(x*(b^2-a^2)+a^4)-SQRT(x)*SQRT(a^2-x)*SQ~
RT(y*(b^2-a^2)+a^4))/((a^2-b^2)*SQRT(x*(b^2-a^2)+a^4)*SQRT(y*(b^2-a^2)+a^4))

;Sub(#61)
AHAT(e1+e2)=(SQRT(y)*SQRT(a^2-y)*SQRT(a^2*(b^2-a^2)+a^4)-SQRT(a^2)*SQRT(a^2-a~
^2)*SQRT(y*(b^2-a^2)+a^4))/((a^2-b^2)*SQRT(a^2*(b^2-a^2)+a^4)*SQRT(y*(b^2-a^2~
)+a^4))

;Simp(#62)
AHAT(e1+e2)=SQRT(y)*SQRT(a^2-y)/((a^2-b^2)*SQRT(y*(b^2-a^2)+a^4))

;Sub(User')
AHAT(e1+e2)=LIM(SQRT(y)*SQRT(a^2-y)/((a^2-b^2)*SQRT(y*(b^2-a^2)+a^4)),y,0,0)

;Simp(#64)
AHAT(e1+e2)=0

"Substitute AHAT(e1 + e2)."

IHAT(e3)=2*AHAT(e1+e2)+r31*r32*IHAT(-e3)

;Simp(User)
IHAT(e3)=r31*r32*IHAT(-e3)

RA(i,j,a,b):=a SUB i/b SUB i-a SUB j/b SUB j

;Sub(#68)
IHAT(e3)=RA(3,1,a,b)*RA(3,2,a,b)*IHAT(-e3)

;Sub(#70)
IHAT(e3)=RA(3,1,[0,a^2,a^4],[1,-1,-(a^2-b^2)])*RA(3,2,[0,a^2,a^4],[1,-1,-(a^2~
-b^2)])*IHAT(-e3)

;Simp(#71)
IHAT(e3)=a^6*b^2*IHAT(-e3)/(a^2-b^2)^2

"Substitute IHAT(e3)."

perimeter=2*(b^2-a^2)*IHAT(e3)/a

;Simp(User)
perimeter=2*a^5*b^2*IHAT(-e3)/(b^2-a^2)

"Find the function to evaluate IHAT(-e3)."

LOAD("c2002.mth")

IHAT(-e3)=ISHAT(IHAT(-e3),[0,a^2,a^4,1],[1,-1,-(a^2-b^2),0],a^2,0)

IHAT(-e3)=(b^2-a^2)*I428([[0,a^2,a^4,1] SUB 3,[0,a^2,a^4,1] SUB 2,[0,a^2,a^4,~
1] SUB 1,[0,a^2,a^4,1] SUB 4],[[1,-1,b^2-a^2,0] SUB 3,[1,-1,b^2-a^2,0] SUB 2,~
[1,-1,b^2-a^2,0] SUB 1,[1,-1,b^2-a^2,0] SUB 4],a^2,0)

;Simp(User')
IHAT(-e3)=(b^2-a^2)*I428([a^4,a^2,0,1],[b^2-a^2,-1,1,0],a^2,0)

"I428(a, b, x, y) is the integral with n=4 and h=4. I428A(a, b, x, y) is the ~
same integral with n=3 and h=3."

"Load the program to evaluate I428A(a, b, x, y)."

LOAD("c1998.mth")

"Substitute for IHAT(-e3)."

perimeter=2*a^5*b^2*I428A([a^4,a^2,0],[b^2-a^2,-1,1],a^2,0)

"Calculate the perimeter of the ellipse."

[a=7.5,b=5]

;Sub(#85)
perimeter=2*(7.5)^5*5^2*I428A([(7.5)^4,(7.5)^2,0],[5^2-(7.5)^2,-1,1],(7.5)^2,~
0)

;Approx(#88)
perimeter=39.66359897322647

PrecisionDigits:=64

NotationDigits:=64

;Approx(#88)
perimeter=39.66359897322647447832915756945768124182520712081625172416815779

"I428(a, b, x, y) could have been used also by adding a factor that is always~
 1."

"Substitute for IHAT(-e3)."

;Sub(#85')
perimeter=2*a^5*b^2*I428([a^4,a^2,0,1],[b^2-a^2,-1,1,0],a^2,0)

[a=7.5,b=5]

;Sub(#95)
perimeter=2*(7.5)^5*5^2*I428([(7.5)^4,(7.5)^2,0,1],[5^2-(7.5)^2,-1,1,0],(7.5)~
^2,0)

;Approx(#97)
perimeter=39.66359897322647447832915756945768124182520712081625172416815779