#1: " ----------[ D e r i v e ]---------- " User #2: " ---------- Initialization ---------- " User #3: " ---------- Determining Zero Equivalence ---------- " User #4: " The following expressions are all equal to zero " User 3 1/6 #5: SQRT(997) - (997 ) User #6: 0 Simp(#5) 3 1/6 #7: SQRT(999983) - (999983 ) User #8: 0 Simp(#7) 1/3 1/3 3 1/3 1/3 #9: (2 + 4 ) - 6*(2 + 4 ) - 6 User #10: 0 Simp(#9) 3 2 #11: COS(x) + COS(x)*SIN(x) - COS(x) User #12: 0 Simp(#11) User #13: " See Joel Moses, ``Algebraic Simplification: A Guide for the Perplexed'', " User #14: " _Communications of the Association of Computing Machinery_, Volume 14, " User #15: " Number 8, August 1971, 527--537. This expression is zero if Re(x) is " User #16: " contained in the interval ((4 n - 1)/2 pi, (4 n + 1)/2 pi) for n an integer: " User #17: " ..., (-5/2 pi, -3/2 pi), (-pi/2, pi/2), (3/2 pi, 5/2 pi), ... " / / 1 pi \\ #18: expr := LOG|TAN|---*x + ----|| - ASINH(TAN(x)) User \ \ 2 4 // #19: expr User Simp(#19) / / 2*x + pi \ \ #20: LN(COS(x)) - LN|COT|----------|*(SIGN(COS(x)) + SIN(x))| \ \ 4 / / #21: Logarithm := Expand User #22: expr User Simp(#22) / SIGN(COS(x)) + SIN(x) \ #23: - LN|-----------------------| + LN(COS(x)) - LN(1 - SIN(x)) \ COS(x) / #24: x :epsilon Real (- pi/2, pi/2) User #25: x Simp(#24) #26: expr User Simp(#26) / SIGN(COS(x)) + SIN(x) \ #27: - LN|-----------------------| + LN(COS(x)) - LN(1 - SIN(x)) \ COS(x) / #28: x := User #29: Logarithm := Auto User User #30: " Use a roundabout method---show that expr is a constant equal to zero " d #31: q_ := -- expr User dx #32: q_ User / 2*x + pi \ COT|----------|*(SIGN(COS(x)) - 1) #33: \ 4 / Simp(#32) ------------------------------------ SIGN(COS(x)) + SIN(x) #34: Trigonometry := Expand User #35: q_ User (1 - SIN(x))*(SIGN(COS(x)) - 1) #36: --------------------------------- Simp(#35) COS(x)*(SIGN(COS(x)) + SIN(x)) #37: x :epsilon Real (- pi/2, pi/2) User #38: x Simp(#37) #39: q_ User (1 - SIN(x))*(SIGN(COS(x)) - 1) #40: --------------------------------- Simp(#39) COS(x)*(SIGN(COS(x)) + SIN(x)) #41: Trigonometry := Auto User #42: x := 0 User #43: expr User #44: 0 Simp(#43) #45: x := User #46: expr := User / 2*SQRT(r) + 1 \ #47: LOG|---------------------------| User \ SQRT(4*r + 4*SQRT(r) + 1) / / 2 \ | SQRT((2*SQRT(r) + 1) ) | #48: LN|------------------------| Simp(#47) \ 2*SQRT(r) + 1 / User SQRT(r)/(2*SQRT(r) + 1) #49: (4*r + 4*SQRT(r) + 1) *(2*SQRT(r) + 1/(2*SQRT(r) + 1) 1) - 2*SQRT(r) - 1 Simp(#49) 1/(2*SQRT(r) + 1) #50: (2*SQRT(r) + 1) *((2*SQRT(r) + 2 SQRT(r)/(2*SQRT(r) + 1) 1) ) - 2*SQRT(r) - 1 #51: r :epsilon Real [0, inf) User #52: r Simp(#51) / 2*SQRT(r) + 1 \ #53: LOG|---------------------------| User \ SQRT(4*r + 4*SQRT(r) + 1) / #54: 0 Simp(#53) User SQRT(r)/(2*SQRT(r) + 1) #55: (4*r + 4*SQRT(r) + 1) *(2*SQRT(r) + 1/(2*SQRT(r) + 1) 1) - 2*SQRT(r) - 1 #56: 0 Simp(#55) #57: r := User #58: " [Gradshteyn and Ryzhik 9.535(3)] " User 1 - z / z*pi \ z #59: 2 *GAMMA(z)*ZETA(z)*COS|------| - pi *ZETA(1 - z) User \ 2 / Simp(#59) 1 - z / pi*z \ z #60: 2 *ZETA(z)*(z - 1)!*COS|------| - pi *ZETA(1 - z) \ 2 / #61: " ---------- Quit ---------- " User