Symbolic algebra and Mathematics with Xcas Renée De Graeve, Bernard Parisse University of Grenoble I 2 c 2002, 2007 Renée De Graeve, Bernard Parisse [email protected] [email protected] Contents 1 Index 17 2 The CAS functions 27 2.1 Symbolic constants : e pi infinity i ............ 27 2.2 Booleans .............................. 27 2.2.1 The values of a boolean : true false ......... 27 2.2.2 Tests : ==, !=, >, >=, <, =< ........... 27 2.2.3 Boolean operators : or xor and not ......... 28 2.2.4 Transform a boolean expression as a list : exp2list . 29 2.2.5 Evaluate booleans : evalb ................ 29 2.3 Operators bit to bit ......................... 30 2.3.1 Operators bitor, bitxor, bitand ......... 30 2.3.2 Hamming distance bit to bit : hamdist .......... 31 2.4 Strings ................................ 31 2.4.1 Character and string : " .................. 31 2.4.2 First character, middle and end of a string : head mid tail ............................ 32 2.4.3 Concatenation of a sequence of words : cumSum ..... 32 2.4.4 ASCII code of a character : ord .............. 33 2.4.5 ASCII code of a string : asc ................ 33 2.4.6 String defined by the ASCII codes of its characters : char 34 2.4.7 Find a character in a string : inString ......... 34 2.4.8 Concat objects into a string : cat ............. 35 2.4.9 Add an object to a string : + ................ 35 2.4.10 Transform an integer into a string : cat + ........ 36 2.4.11 Transform a string into a number : expr ......... 36 2.5 Write an integer in a b basis: convert .............. 37 2.6 Integers (and Gaussian Integers) .................. 38 2.6.1 The factorial : factorial ................ 38 2.6.2 GCD : gcd igcd ..................... 39 2.6.3 GCD : Gcd ......................... 40 2.6.4 GCD of a list of integers : lgcd .............. 40 2.6.5 The least common multiple : lcm ............. 41 2.6.6 Decomposition into prime factors : ifactor ....... 41 2.6.7 List of prime factors : ifactors ............. 41 2.6.8 Matrix of factors : maple_ifactors .......... 42 2.6.9 The divisors of a number : idivis divisors ..... 42 3 4 CONTENTS 2.6.10 The integer Euclidean quotient : iquo intDiv ..... 42 2.6.11 The integer Euclidean remainder : irem remain smod mods mod % ....................... 43 2.6.12 Euclidean quotient and euclidean remainder of two inte- gers : iquorem ...................... 44 2.6.13 Test of evenness : even .................. 44 2.6.14 Test of oddness : odd ................... 45 2.6.15 Test of pseudo-primality : is_pseudoprime ...... 45 2.6.16 Test of primality : is_prime isprime isPrime . 46 2.6.17 The smallest pseudo-prime greater than n : nextprime . 47 2.6.18 The greatest pseudo-prime less than n : prevprime . 47 2.6.19 The n-th prime number : ithprime ........... 47 2.6.20 Bézout’s Identity : iegcd igcdex ........... 48 2.6.21 Solving au+bv=c in Z: iabcuv .............. 48 2.6.22 Chinese remainders : ichinrem, ichrem ....... 48 2.6.23 Chinese remainders for lists of integers : chrem ..... 50 2.6.24 Solving a2 + b2 = p in Z : pa2b2 ............ 51 2.6.25 The Euler indicatrix : euler phi ............ 51 2.6.26 Legendre symbol : legendre_symbol ......... 51 2.6.27 Jacobi symbol : jacobi_symbol ............ 52 2.7 Combinatory analysis ........................ 53 2.7.1 Factorial : factorial ! ................ 53 2.7.2 Binomial coefficients : binomial comb nCr ..... 53 2.7.3 Arrangements : perm nPr ................ 54 2.7.4 Random integers : rand .................. 54 2.8 Rationals .............................. 54 2.8.1 Transform a floating point number into a rational : exact float2rational .................... 54 2.8.2 Integer and fractional part : propfrac propFrac . 55 2.8.3 Numerator of a fraction after simplification : numer getNum .......................... 56 2.8.4 Denominator of a fraction after simplification : denom getDenom ......................... 56 2.8.5 Numerator and denominator of a fraction : f2nd fxnd . 57 2.8.6 Simplification of a pair of integers : simp2 ........ 57 2.8.7 Continued fraction representation of a real : dfc ..... 57 2.8.8 Transform a continued fraction representation into a real : dfc2f ........................... 59 2.8.9 The n-th Bernoulli number : bernoulli ........ 61 2.8.10 Access to PARI/GP commands: pari ........... 61 2.9 Real numbers ............................ 61 2.9.1 Eval a real at a given precision : evalf and Digits, DIGITS) .......................... 61 2.9.2 Usual infixed functions on reals : +,-,*,/,ˆ ..... 63 2.9.3 Usual prefixed functions on reals : rdiv ......... 64 2.9.4 n-th root : root ...................... 65 2.9.5 Error function : erf .................... 65 2.9.6 Complementary error function: erfc ........... 66 CONTENTS 5 2.9.7 The Γ function : Gamma .................. 67 2.9.8 The β function : Beta ................... 68 2.9.9 Derivatives of the DiGamma fonction : Psi ........ 68 2.9.10 The ζ function : Zeta ................... 69 2.9.11 Airy functions : Airy_Ai and Airy_Bi ......... 69 2.10 Permutations ............................ 70 2.10.1 Random permutation : randperm ............ 71 2.10.2 Decomposition as a product of disjoint cycles : permu2cycles ...................... 71 2.10.3 Product of disjoint cycles to permutation: cycles2permu 71 2.10.4 Transform a cycle into permutation : cycle2perm . 72 2.10.5 Transform a permutation into a matrix : permu2mat . 72 2.10.6 Checking for a permutation : is_permu ......... 72 2.10.7 Checking for a cycle : is_cycle ............. 73 2.10.8 Product of two permutations : p1op2 ........... 73 2.10.9 Composition of a cycle and a permutation : c1op2 . 74 2.10.10 Composition of a permutation and a cycle : p1oc2 . 74 2.10.11 Product of two cycles : c1oc2 .............. 74 2.10.12 Signature of a permutation : signature ......... 75 2.10.13 Inverse of a permutation : perminv ........... 75 2.10.14 Inverse of a cycle : cycleinv .............. 75 2.10.15 Order of a permutation : permuorder .......... 75 2.10.16 Group generated by two permutations : groupermu . 76 2.11 Complex numbers .......................... 76 2.11.1 Usual complex functions : +,-,*,/,ˆ ......... 76 2.11.2 Real part of a complex number : re real ........ 76 2.11.3 Imaginary part of a complex number : im imag ..... 77 2.11.4 Write a complex as re(z)+i*im(z) : evalc ..... 77 2.11.5 Modulus of a complex number : abs ........... 77 2.11.6 Argument of a complex number : arg ........... 77 2.11.7 The normalized complex number : normalize unitV 78 2.11.8 Conjuguate of a complex number : conj ......... 78 2.11.9 Multiplication by the complex conjugate : mult_c_conjugate .................. 78 2.11.10 Barycenter of complex numbers : barycentre ..... 79 2.12 Algebraic expressions ........................ 79 2.12.1 Evaluate an expression : eval ............... 79 2.12.2 Evaluate algebraic expressions : evala .......... 80 2.12.3 Prevent evaluation : quote hold ’ ........... 80 2.12.4 Force evaluation : unquote ................ 80 2.12.5 Distributivity : expand fdistrib ........... 80 2.12.6 Canonical form : canonical_form ........... 81 2.12.7 Multiplication by the conjugate quantity : mult_conjugate .................... 81 2.12.8 Separation of variables : split .............. 82 2.12.9 Factorisation : factor .................. 82 2.12.10 Complex factorisation : cFactor ............. 84 2.12.11 Zeros of an expression : zeros .............. 84 6 CONTENTS 2.12.12 Complex zeros of an expression : cZeros ........ 85 2.12.13 Normal form : normal .................. 86 2.12.14 Simplify : simplify ................... 86 2.12.15 Normal form for rational fractions : ratnormal ..... 87 2.12.16 Substitue a variable by a value : subst .......... 87 2.12.17 Substitue a variable by a value (Maple and Mupad compat- ibility) : subs ....................... 89 2.12.18 Evaluate a primitive at boundaries: preval ....... 90 2.12.19 Sub-expression of an expression : part .......... 90 2.13 Values of un ............................. 91 2.13.1 Array of values of a sequence : tablefunc ....... 91 2.13.2 Table of values and graph of a recurrent sequence : tableseq and plotseq ....................... 91 2.14 Operators or infixed functions ................... 92 2.14.1 Usual operators :+, -, *, /, ˆ ........... 92 2.14.2 Xcas operators ...................... 92 2.14.3 Define an operator: user_operator .......... 93 2.15 Functions and expressions with symbolic variables ........ 94 2.15.1 Difference between function and expression ........ 94 2.15.2 Transform an expression into a fonction : unapply . 94 2.15.3 Top and leaves of an expression : sommet feuille op 95 2.16 Functions .............................. 97 2.16.1 Context-dependant functions. ............... 97 2.16.2 Usual functions ....................... 98 2.16.3 Defining algebraic functions ................ 99 2.16.4 Composition of two functions: @ .............. 101 2.16.5 Repeted function composition: @@ ............. 102 2.16.6 Define a fonction with the history : as_function_of . 102 2.17 Derivation and applications. .................... 104 2.17.1 Functional derivative : function_diff ........ 104 2.17.2 Length of an arc : arcLen ................ 105 2.17.3 Maximum and minimum of an expression: fMax fMin . 106 2.17.4 Table of values and graph : tablefunc and plotfunc 106 2.17.5 Derivative and partial derivative .............. 107 2.18 Integration .............................. 109 2.18.1 Antiderivative and definite integral : integrate int Int ............................. 109 2.18.2 Discrete summation: sum ................. 111 2.18.3 Riemann sum : sum_riemann