Coverage for local/lib/python2.7/site-packages/sage/calculus/wester.py : 100%

r""" 

Further examples from Wester's paper 

These are all the problems at 

http://yacas.sourceforge.net/essaysmanual.html 

They come from the 1994 paper "Review of CAS mathematical 

capabilities", by Michael Wester, who put forward 123 problems that 

a reasonable computer algebra system should be able to solve and 

tested the then current versions of various commercial CAS on this 

list. Sage can do most of the problems natively now, i.e., with no 

explicit calls to Maxima or other systems. 

:: 

sage: # (YES) factorial of 50, and factor it 

sage: factorial(50) 

30414093201713378043612608166064768844377641568960512000000000000 

sage: factor(factorial(50)) 

2^47 * 3^22 * 5^12 * 7^8 * 11^4 * 13^3 * 17^2 * 19^2 * 23^2 * 29 * 31 * 37 * 41 * 43 * 47 

:: 

sage: # (YES) 1/2+...+1/10 = 4861/2520 

sage: sum(1/n for n in range(2,10+1)) == 4861/2520 

True 

:: 

sage: # (YES) Evaluate e^(Pi*Sqrt(163)) to 50 decimal digits 

sage: a = e^(pi*sqrt(163)); a 

e^(sqrt(163)*pi) 

sage: RealField(150)(a) 

2.6253741264076874399999999999925007259719820e17 

:: 

sage: # (YES) Evaluate the Bessel function J[2] numerically at z=1+I. 

sage: bessel_J(2, 1+I).n() 

0.0415798869439621 + 0.247397641513306*I 

:: 

sage: # (YES) Obtain period of decimal fraction 1/7=0.(142857). 

sage: a = 1/7 

sage: a 

1/7 

sage: a.period() 

6 

:: 

sage: # (YES) Continued fraction of 3.1415926535 

sage: a = 3.1415926535 

sage: continued_fraction(a) 

[3; 7, 15, 1, 292, 1, 1, 6, 2, 13, 4] 

:: 

sage: # (YES) Sqrt(2*Sqrt(3)+4)=1+Sqrt(3). 

sage: # The Maxima backend equality checker does this; 

sage: # note the equality only holds for one choice of sign, 

sage: # but Maxima always chooses the "positive" one 

sage: a = sqrt(2*sqrt(3) + 4); b = 1 + sqrt(3) 

sage: float(a-b) 

0.0 

sage: bool(a == b) 

True 

sage: # We can, of course, do this in a quadratic field 

sage: k.<sqrt3> = QuadraticField(3) 

sage: asqr = 2*sqrt3 + 4 

sage: b = 1+sqrt3 

sage: asqr == b^2 

True 

:: 

sage: # (YES) Sqrt(14+3*Sqrt(3+2*Sqrt(5-12*Sqrt(3-2*Sqrt(2)))))=3+Sqrt(2). 

sage: a = sqrt(14+3*sqrt(3+2*sqrt(5-12*sqrt(3-2*sqrt(2))))) 

sage: b = 3+sqrt(2) 

sage: a, b 

(sqrt(3*sqrt(2*sqrt(-12*sqrt(-2*sqrt(2) + 3) + 5) + 3) + 14), sqrt(2) + 3) 

sage: bool(a==b) 

True 

sage: abs(float(a-b)) < 1e-10 

True 

sage: # 2*Infinity-3=Infinity. 

sage: 2*infinity-3 == infinity 

True 

:: 

sage: # (YES) Standard deviation of the sample (1, 2, 3, 4, 5). 

sage: v = vector(RDF, 5, [1,2,3,4,5]) 

sage: v.standard_deviation() 

1.5811388300841898 

:: 

sage: # (NO) Hypothesis testing with t-distribution. 

sage: # (NO) Hypothesis testing with chi^2 distribution 

sage: # (But both are included in Scipy and R) 

:: 

sage: # (YES) (x^2-4)/(x^2+4*x+4)=(x-2)/(x+2). 

sage: R.<x> = QQ[] 

sage: (x^2-4)/(x^2+4*x+4) == (x-2)/(x+2) 

True 

sage: restore('x') 

:: 

sage: # (YES -- Maxima doesn't immediately consider them 

sage: # equal, but simplification shows that they are) 

sage: # (Exp(x)-1)/(Exp(x/2)+1)=Exp(x/2)-1. 

sage: f = (exp(x)-1)/(exp(x/2)+1) 

sage: g = exp(x/2)-1 

sage: f 

(e^x - 1)/(e^(1/2*x) + 1) 

sage: g 

e^(1/2*x) - 1 

sage: f.canonicalize_radical() 

e^(1/2*x) - 1 

sage: g 

e^(1/2*x) - 1 

sage: f(x=10.0).n(53), g(x=10.0).n(53) 

(147.413159102577, 147.413159102577) 

sage: bool(f == g) 

True 

:: 

sage: # (YES) Expand (1+x)^20, take derivative and factorize. 

sage: # first do it using algebraic polys 

sage: R.<x> = QQ[] 

sage: f = (1+x)^20; f 

x^20 + 20*x^19 + 190*x^18 + 1140*x^17 + 4845*x^16 + 15504*x^15 + 38760*x^14 + 77520*x^13 + 125970*x^12 + 167960*x^11 + 184756*x^10 + 167960*x^9 + 125970*x^8 + 77520*x^7 + 38760*x^6 + 15504*x^5 + 4845*x^4 + 1140*x^3 + 190*x^2 + 20*x + 1 

sage: deriv = f.derivative() 

sage: deriv 

20*x^19 + 380*x^18 + 3420*x^17 + 19380*x^16 + 77520*x^15 + 232560*x^14 + 542640*x^13 + 1007760*x^12 + 1511640*x^11 + 1847560*x^10 + 1847560*x^9 + 1511640*x^8 + 1007760*x^7 + 542640*x^6 + 232560*x^5 + 77520*x^4 + 19380*x^3 + 3420*x^2 + 380*x + 20 

sage: deriv.factor() 

(20) * (x + 1)^19 

sage: restore('x') 

sage: # next do it symbolically 

sage: var('y') 

y 

sage: f = (1+y)^20; f 

(y + 1)^20 

sage: g = f.expand(); g 

y^20 + 20*y^19 + 190*y^18 + 1140*y^17 + 4845*y^16 + 15504*y^15 + 38760*y^14 + 77520*y^13 + 125970*y^12 + 167960*y^11 + 184756*y^10 + 167960*y^9 + 125970*y^8 + 77520*y^7 + 38760*y^6 + 15504*y^5 + 4845*y^4 + 1140*y^3 + 190*y^2 + 20*y + 1 

sage: deriv = g.derivative(); deriv 

20*y^19 + 380*y^18 + 3420*y^17 + 19380*y^16 + 77520*y^15 + 232560*y^14 + 542640*y^13 + 1007760*y^12 + 1511640*y^11 + 1847560*y^10 + 1847560*y^9 + 1511640*y^8 + 1007760*y^7 + 542640*y^6 + 232560*y^5 + 77520*y^4 + 19380*y^3 + 3420*y^2 + 380*y + 20 

sage: deriv.factor() 

20*(y + 1)^19 

:: 

sage: # (YES) Factorize x^100-1. 

sage: factor(x^100-1) 

(x^40 - x^30 + x^20 - x^10 + 1)*(x^20 + x^15 + x^10 + x^5 + 1)*(x^20 - x^15 + x^10 - x^5 + 1)*(x^8 - x^6 + x^4 - x^2 + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^4 - x^3 + x^2 - x + 1)*(x^2 + 1)*(x + 1)*(x - 1) 

sage: # Also, algebraically 

sage: x = polygen(QQ) 

sage: factor(x^100 - 1) 

(x - 1) * (x + 1) * (x^2 + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1) * (x^8 - x^6 + x^4 - x^2 + 1) * (x^20 - x^15 + x^10 - x^5 + 1) * (x^20 + x^15 + x^10 + x^5 + 1) * (x^40 - x^30 + x^20 - x^10 + 1) 

sage: restore('x') 

:: 

sage: # (YES) Factorize x^4-3*x^2+1 in the field of rational numbers extended by roots of x^2-x-1. 

sage: k.< a> = NumberField(x^2 - x -1) 

sage: R.< y> = k[] 

sage: f = y^4 - 3*y^2 + 1 

sage: f 

y^4 - 3*y^2 + 1 

sage: factor(f) 

(y - a) * (y - a + 1) * (y + a - 1) * (y + a) 

:: 

sage: # (YES) Factorize x^4-3*x^2+1 mod 5. 

sage: k.< x > = GF(5) [ ] 

sage: f = x^4 - 3*x^2 + 1 

sage: f.factor() 

(x + 2)^2 * (x + 3)^2 

sage: # Alternatively, from symbol x as follows: 

sage: reset('x') 

sage: f = x^4 - 3*x^2 + 1 

sage: f.polynomial(GF(5)).factor() 

(x + 2)^2 * (x + 3)^2 

:: 

sage: # (YES) Partial fraction decomposition of (x^2+2*x+3)/(x^3+4*x^2+5*x+2) 

sage: f = (x^2+2*x+3)/(x^3+4*x^2+5*x+2); f 

(x^2 + 2*x + 3)/(x^3 + 4*x^2 + 5*x + 2) 

sage: f.partial_fraction() 

3/(x + 2) - 2/(x + 1) + 2/(x + 1)^2 

:: 

sage: # (YES) Assuming x>=y, y>=z, z>=x, deduce x=z. 

sage: forget() 

sage: var('x,y,z') 

(x, y, z) 

sage: assume(x>=y, y>=z,z>=x) 

sage: bool(x==z) 

True 

:: 

sage: # (YES) Assuming x>y, y>0, deduce 2*x^2>2*y^2. 

sage: forget() 

sage: assume(x>y, y>0) 

sage: list(sorted(assumptions())) 

[x > y, y > 0] 

sage: bool(2*x^2 > 2*y^2) 

True 

sage: forget() 

sage: assumptions() 

[] 

:: 

sage: # (NO) Solve the inequality Abs(x-1)>2. 

sage: # Maxima doesn't solve inequalities 

sage: # (but some Maxima packages do): 

sage: eqn = abs(x-1) > 2 

sage: eqn 

abs(x - 1) > 2 

:: 

sage: # (NO) Solve the inequality (x-1)*...*(x-5)<0. 

sage: eqn = prod(x-i for i in range(1,5 +1)) < 0 

sage: # but don't know how to solve 

sage: eqn 

(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) < 0 

:: 

sage: # (YES) Cos(3*x)/Cos(x)=Cos(x)^2-3*Sin(x)^2 or similar equivalent combination. 

sage: f = cos(3*x)/cos(x) 

sage: g = cos(x)^2 - 3*sin(x)^2 

sage: h = f-g 

sage: h.trig_simplify() 

0 

:: 

sage: # (YES) Cos(3*x)/Cos(x)=2*Cos(2*x)-1. 

sage: f = cos(3*x)/cos(x) 

sage: g = 2*cos(2*x) - 1 

sage: h = f-g 

sage: h.trig_simplify() 

0 

:: 

sage: # (GOOD ENOUGH) Define rewrite rules to match Cos(3*x)/Cos(x)=Cos(x)^2-3*Sin(x)^2. 

sage: # Sage has no notion of "rewrite rules", but 

sage: # it can simplify both to the same thing. 

sage: (cos(3*x)/cos(x)).simplify_full() 

4*cos(x)^2 - 3 

sage: (cos(x)^2-3*sin(x)^2).simplify_full() 

4*cos(x)^2 - 3 

:: 

sage: # (YES) Sqrt(997)-(997^3)^(1/6)=0 

sage: a = sqrt(997) - (997^3)^(1/6) 

sage: a.simplify() 

0 

sage: bool(a == 0) 

True 

:: 

sage: # (YES) Sqrt(99983)-99983^3^(1/6)=0 

sage: a = sqrt(99983) - (99983^3)^(1/6) 

sage: bool(a==0) 

True 

sage: float(a) 

1.1368683772...e-13 

sage: 13*7691 

99983 

:: 

sage: # (YES) (2^(1/3) + 4^(1/3))^3 - 6*(2^(1/3) + 4^(1/3))-6 = 0 

sage: a = (2^(1/3) + 4^(1/3))^3 - 6*(2^(1/3) + 4^(1/3)) - 6; a 

(4^(1/3) + 2^(1/3))^3 - 6*4^(1/3) - 6*2^(1/3) - 6 

sage: bool(a==0) 

True 

sage: abs(float(a)) < 1e-10 

True 

sage: ## or we can do it using number fields. 

sage: reset('x') 

sage: k.<b> = NumberField(x^3-2) 

sage: a = (b + b^2)^3 - 6*(b + b^2) - 6 

sage: a 

0 

:: 

sage: # (NO, except numerically) Ln(Tan(x/2+Pi/4))-ArcSinh(Tan(x))=0 

# Sage uses the Maxima convention when comparing symbolic expressions and 

# returns True only when it can prove equality. Thus, in this case, we get 

# False even though the equality holds. 

sage: f = log(tan(x/2 + pi/4)) - arcsinh(tan(x)) 

sage: bool(f == 0) 

False 

sage: [abs(float(f(x=i/10))) < 1e-15 for i in range(1,5)] 

[True, True, True, True] 

sage: # Numerically, the expression Ln(Tan(x/2+Pi/4))-ArcSinh(Tan(x))=0 and its derivative at x=0 are zero. 

sage: g = f.derivative() 

sage: abs(float(f(x=0))) < 1e-10 

True 

sage: abs(float(g(x=0))) < 1e-10 

True 

sage: g 

-sqrt(tan(x)^2 + 1) + 1/2*(tan(1/4*pi + 1/2*x)^2 + 1)/tan(1/4*pi + 1/2*x) 

:: 

sage: # (NO) Ln((2*Sqrt(r) + 1)/Sqrt(4*r 4*Sqrt(r) 1))=0. 

sage: var('r') 

r 

sage: f = log( (2*sqrt(r) + 1) / sqrt(4*r + 4*sqrt(r) + 1)) 

sage: f 

log((2*sqrt(r) + 1)/sqrt(4*r + 4*sqrt(r) + 1)) 

sage: bool(f == 0) 

False 

sage: [abs(float(f(r=i))) < 1e-10 for i in [0.1,0.3,0.5]] 

[True, True, True] 

:: 

sage: # (NO) 

sage: # (4*r+4*Sqrt(r)+1)^(Sqrt(r)/(2*Sqrt(r)+1))*(2*Sqrt(r)+1)^(2*Sqrt(r)+1)^(-1)-2*Sqrt(r)-1=0, assuming r>0. 

sage: assume(r>0) 

sage: f = (4*r+4*sqrt(r)+1)^(sqrt(r)/(2*sqrt(r)+1))*(2*sqrt(r)+1)^(2*sqrt(r)+1)^(-1)-2*sqrt(r)-1 

sage: f 

(4*r + 4*sqrt(r) + 1)^(sqrt(r)/(2*sqrt(r) + 1))*(2*sqrt(r) + 1)^(1/(2*sqrt(r) + 1)) - 2*sqrt(r) - 1 

sage: bool(f == 0) 

False 

sage: [abs(float(f(r=i))) < 1e-10 for i in [0.1,0.3,0.5]] 

[True, True, True] 

:: 

sage: # (YES) Obtain real and imaginary parts of Ln(3+4*I). 

sage: a = log(3+4*I); a 

log(4*I + 3) 

sage: a.real() 

log(5) 

sage: a.imag() 

arctan(4/3) 

:: 

sage: # (YES) Obtain real and imaginary parts of Tan(x+I*y) 

sage: z = var('z') 

sage: a = tan(z); a 

tan(z) 

sage: a.real() 

sin(2*real_part(z))/(cos(2*real_part(z)) + cosh(2*imag_part(z))) 

sage: a.imag() 

sinh(2*imag_part(z))/(cos(2*real_part(z)) + cosh(2*imag_part(z))) 

:: 

sage: # (YES) Simplify Ln(Exp(z)) to z for -Pi<Im(z)<=Pi. 

sage: # Unfortunately (?), Maxima does this even without 

sage: # any assumptions. 

sage: # We *would* use assume(-pi < imag(z)) 

sage: # and assume(imag(z) <= pi) 

sage: f = log(exp(z)); f 

log(e^z) 

sage: f.simplify() 

z 

sage: forget() 

:: 

sage: # (YES) Assuming Re(x)>0, Re(y)>0, deduce x^(1/n)*y^(1/n)-(x*y)^(1/n)=0. 

sage: # Maxima 5.26 has different behaviours depending on the current 

sage: # domain. 

sage: # To stick with the behaviour of previous versions, the domain is set 

sage: # to 'real' in the following. 

sage: # See Trac #10682 for further details. 

sage: n = var('n') 

sage: f = x^(1/n)*y^(1/n)-(x*y)^(1/n) 

sage: assume(real(x) > 0, real(y) > 0) 

sage: f.simplify() 

x^(1/n)*y^(1/n) - (x*y)^(1/n) 

sage: maxima = sage.calculus.calculus.maxima 

sage: maxima.set('domain', 'real') # set domain to real 

sage: f.simplify() 

0 

sage: maxima.set('domain', 'complex') # set domain back to its default value 

sage: forget() 

:: 

sage: # (YES) Transform equations, (x==2)/2+(1==1)=>x/2+1==2. 

sage: eq1 = x == 2 

sage: eq2 = SR(1) == SR(1) 

sage: eq1/2 + eq2 

1/2*x + 1 == 2 

:: 

sage: # (SOMEWHAT) Solve Exp(x)=1 and get all solutions. 

sage: # to_poly_solve in Maxima can do this. 

sage: solve(exp(x) == 1, x) 

[x == 0] 

:: 

sage: # (SOMEWHAT) Solve Tan(x)=1 and get all solutions. 

sage: # to_poly_solve in Maxima can do this. 

sage: solve(tan(x) == 1, x) 

[x == 1/4*pi] 

:: 

sage: # (YES) Solve a degenerate 3x3 linear system. 

sage: # x+y+z==6,2*x+y+2*z==10,x+3*y+z==10 

sage: # First symbolically: 

sage: solve([x+y+z==6, 2*x+y+2*z==10, x+3*y+z==10], x,y,z) 

[[x == -r1 + 4, y == 2, z == r1]] 

:: 

sage: # (YES) Invert a 2x2 symbolic matrix. 

sage: # [[a,b],[1,a*b]] 

sage: # Using multivariate poly ring -- much nicer 

sage: R.<a,b> = QQ[] 

sage: m = matrix(2,2,[a,b, 1, a*b]) 

sage: zz = m^(-1) 

sage: zz 

[ a/(a^2 - 1) (-1)/(a^2 - 1)] 

[(-1)/(a^2*b - b) a/(a^2*b - b)] 

:: 

sage: # (YES) Compute and factor the determinant of the 4x4 Vandermonde matrix in a, b, c, d. 

sage: var('a,b,c,d') 

(a, b, c, d) 

sage: m = matrix(SR, 4, 4, [[z^i for i in range(4)] for z in [a,b,c,d]]) 

sage: m 

[ 1 a a^2 a^3] 

[ 1 b b^2 b^3] 

[ 1 c c^2 c^3] 

[ 1 d d^2 d^3] 

sage: d = m.determinant() 

sage: d.factor() 

(a - b)*(a - c)*(a - d)*(b - c)*(b - d)*(c - d) 

:: 

sage: # (YES) Compute and factor the determinant of the 4x4 Vandermonde matrix in a, b, c, d. 

sage: # Do it instead in a multivariate ring 

sage: R.<a,b,c,d> = QQ[] 

sage: m = matrix(R, 4, 4, [[z^i for i in range(4)] for z in [a,b,c,d]]) 

sage: m 

[ 1 a a^2 a^3] 

[ 1 b b^2 b^3] 

[ 1 c c^2 c^3] 

[ 1 d d^2 d^3] 

sage: d = m.determinant() 

sage: d 

a^3*b^2*c - a^2*b^3*c - a^3*b*c^2 + a*b^3*c^2 + a^2*b*c^3 - a*b^2*c^3 - a^3*b^2*d + a^2*b^3*d + a^3*c^2*d - b^3*c^2*d - a^2*c^3*d + b^2*c^3*d + a^3*b*d^2 - a*b^3*d^2 - a^3*c*d^2 + b^3*c*d^2 + a*c^3*d^2 - b*c^3*d^2 - a^2*b*d^3 + a*b^2*d^3 + a^2*c*d^3 - b^2*c*d^3 - a*c^2*d^3 + b*c^2*d^3 

sage: d.factor() 

(-1) * (c - d) * (-b + c) * (b - d) * (-a + c) * (-a + b) * (a - d) 

:: 

sage: # (YES) Find the eigenvalues of a 3x3 integer matrix. 

sage: m = matrix(QQ, 3, [5,-3,-7, -2,1,2, 2,-3,-4]) 

sage: m.eigenspaces_left() 

[ 

(3, Vector space of degree 3 and dimension 1 over Rational Field 

User basis matrix: 

[ 1 0 -1]), 

(1, Vector space of degree 3 and dimension 1 over Rational Field 

User basis matrix: 

[ 1 1 -1]), 

(-2, Vector space of degree 3 and dimension 1 over Rational Field 

User basis matrix: 

[0 1 1]) 

] 

:: 

sage: # (YES) Verify some standard limits found by L'Hopital's rule: 

sage: # Verify(Limit(x,Infinity) (1+1/x)^x, Exp(1)); 

sage: # Verify(Limit(x,0) (1-Cos(x))/x^2, 1/2); 

sage: limit( (1+1/x)^x, x = oo) 

e 

sage: limit( (1-cos(x))/(x^2), x = 1/2) 

-4*cos(1/2) + 4 

:: 

sage: # (OK-ish) D(x)Abs(x) 

sage: # Verify(D(x) Abs(x), Sign(x)); 

sage: diff(abs(x)) 

1/2*(x + conjugate(x))/abs(x) 

sage: _.simplify_full() 

x/abs(x) 

sage: _ = var('x', domain='real') 

sage: diff(abs(x)) 

x/abs(x) 

sage: forget() 

:: 

sage: # (YES) (Integrate(x)Abs(x))=Abs(x)*x/2 

sage: integral(abs(x), x) 

1/2*x*abs(x) 

:: 

sage: # (YES) Compute derivative of Abs(x), piecewise defined. 

sage: # Verify(D(x)if(x<0) (-x) else x, 

sage: # Simplify(if(x<0) -1 else 1)) 

Piecewise defined function with 2 parts, [[(-10, 0), -1], [(0, 10), 1]] 

sage: # (NOT really) Integrate Abs(x), piecewise defined. 

sage: # Verify(Simplify(Integrate(x) 

sage: # if(x<0) (-x) else x), 

sage: # Simplify(if(x<0) (-x^2/2) else x^2/2)); 

sage: f = piecewise([ ((-10,0), -x), ((0,10), x)]) 

sage: f.integral(definite=True) 

100 

:: 

sage: # (YES) Taylor series of 1/Sqrt(1-v^2/c^2) at v=0. 

sage: var('v,c') 

(v, c) 

sage: taylor(1/sqrt(1-v^2/c^2), v, 0, 7) 

1/2*v^2/c^2 + 3/8*v^4/c^4 + 5/16*v^6/c^6 + 1 

:: 

sage: # (OK-ish) (Taylor expansion of Sin(x))/(Taylor expansion of Cos(x)) = (Taylor expansion of Tan(x)). 

sage: # TestYacas(Taylor(x,0,5)(Taylor(x,0,5)Sin(x))/ 

sage: # (Taylor(x,0,5)Cos(x)), Taylor(x,0,5)Tan(x)); 

sage: f = taylor(sin(x), x, 0, 8) 

sage: g = taylor(cos(x), x, 0, 8) 

sage: h = taylor(tan(x), x, 0, 8) 

sage: f = f.power_series(QQ) 

sage: g = g.power_series(QQ) 

sage: h = h.power_series(QQ) 

sage: f - g*h 

O(x^8) 

:: 

sage: # (YES) Taylor expansion of Ln(x)^a*Exp(-b*x) at x=1. 

sage: a,b = var('a,b') 

sage: taylor(log(x)^a*exp(-b*x), x, 1, 3) 

-1/48*(a^3*(x - 1)^a + a^2*(6*b + 5)*(x - 1)^a + 8*b^3*(x - 1)^a + 2*(6*b^2 + 5*b + 3)*a*(x - 1)^a)*(x - 1)^3*e^(-b) + 1/24*(3*a^2*(x - 1)^a + a*(12*b + 5)*(x - 1)^a + 12*b^2*(x - 1)^a)*(x - 1)^2*e^(-b) - 1/2*(a*(x - 1)^a + 2*b*(x - 1)^a)*(x - 1)*e^(-b) + (x - 1)^a*e^(-b) 

:: 

sage: # (YES) Taylor expansion of Ln(Sin(x)/x) at x=0. 

sage: taylor(log(sin(x)/x), x, 0, 10) 

-1/467775*x^10 - 1/37800*x^8 - 1/2835*x^6 - 1/180*x^4 - 1/6*x^2 

:: 

sage: # (NO) Compute n-th term of the Taylor series of Ln(Sin(x)/x) at x=0. 

sage: # need formal functions 

:: 

sage: # (NO) Compute n-th term of the Taylor series of Exp(-x)*Sin(x) at x=0. 

sage: # (Sort of, with some work) 

sage: # Solve x=Sin(y)+Cos(y) for y as Taylor series in x at x=1. 

sage: # TestYacas(InverseTaylor(y,0,4) Sin(y)+Cos(y), 

sage: # (y-1)+(y-1)^2/2+2*(y-1)^3/3+(y-1)^4); 

sage: # Note that InverseTaylor does not give the series in terms of x but in terms of y which is semantically 

sage: # wrong. But other CAS do the same. 

sage: f = sin(y) + cos(y) 

sage: g = f.taylor(y, 0, 10) 

sage: h = g.power_series(QQ) 

sage: k = (h - 1).reverse() 

sage: k 

y + 1/2*y^2 + 2/3*y^3 + y^4 + 17/10*y^5 + 37/12*y^6 + 41/7*y^7 + 23/2*y^8 + 1667/72*y^9 + 3803/80*y^10 + O(y^11) 

:: 

sage: # (OK) Compute Legendre polynomials directly from Rodrigues's formula, P[n]=1/(2^n*n!) *(Deriv(x,n)(x^2-1)^n). 

sage: # P(n,x) := Simplify( 1/(2*n)!! * 

sage: # Deriv(x,n) (x^2-1)^n ); 

sage: # TestYacas(P(4,x), (35*x^4)/8+(-15*x^2)/4+3/8); 

sage: P = lambda n, x: simplify(diff((x^2-1)^n,x,n) / (2^n * factorial(n))) 

sage: P(4,x).expand() 

35/8*x^4 - 15/4*x^2 + 3/8 

:: 

sage: # (YES) Define the polynomial p=Sum(i,1,5,a[i]*x^i). 

sage: # symbolically 

sage: ps = sum(var('a%s'%i)*x^i for i in range(1,6)); ps 

a5*x^5 + a4*x^4 + a3*x^3 + a2*x^2 + a1*x 

sage: ps.parent() 

Symbolic Ring 

sage: # algebraically 

sage: R = PolynomialRing(QQ,5,names='a') 

sage: S.<x> = PolynomialRing(R) 

sage: p = S(list(R.gens()))*x; p 

a4*x^5 + a3*x^4 + a2*x^3 + a1*x^2 + a0*x 

sage: p.parent() 

Univariate Polynomial Ring in x over Multivariate Polynomial Ring in a0, a1, a2, a3, a4 over Rational Field 

:: 

sage: # (YES) Convert the above to Horner's form. 

sage: # Verify(Horner(p, x), ((((a[5]*x+a[4])*x 

sage: # +a[3])*x+a[2])*x+a[1])*x); 

sage: restore('x') 

sage: SR(p).horner(x) 

((((a4*x + a3)*x + a2)*x + a1)*x + a0)*x 

:: 

sage: # (NO) Convert the result of problem 127 to Fortran syntax. 

sage: # CForm(Horner(p, x)); 

:: 

sage: # (YES) Verify that True And False=False. 

sage: (True and False) is False 

True 

:: 

sage: # (YES) Prove x Or Not x. 

sage: for x in [True, False]: 

....: print(x or (not x)) 

True 

True 

:: 

sage: # (YES) Prove x Or y Or x And y=>x Or y. 

sage: for x in [True, False]: 

....: for y in [True, False]: 

....: if x or y or x and y: 

....: if not (x or y): 

....: print("failed!") 

"""