## 28 Sums, Products, and Series ¶

### 28.1 Functions and Variables for Sums and Products ¶

Function: bashindices (expr) ¶

  Transforms the expression expr by giving each summation and product a unique index. This gives `changevar` greater precision when it is working with summations or products. The form of the unique index is `jnumber`. The quantity number is determined by referring to `gensumnum`, which can be changed by the user. For example, `gensumnum:0$` resets it.

  Categories: Sums and products ·

Function: lsum (expr, x, L) ¶

  Represents the sum of expr for each element x in L. A noun form `'lsum` is returned if the argument L does not evaluate to a list.

  Examples:

      (%i1) lsum (x^i, i, [1, 2, 7]);
                                  7    2
      (%o1)                      x  + x  + x
      (%i2) lsum (i^2, i, rootsof (x^3 - 1, x));

                           ====
                           \      2
      (%o2)                 >    i
                           /
                           ====
                                         3
                           i in rootsof(x  - 1, x)

  Categories: Sums and products ·

Function: intosum (expr) ¶

  Moves multiplicative factors outside a summation to inside. If the index is used in the outside expression, then the function tries to find a reasonable index, the same as it does for `sumcontract`. This is essentially the reverse idea of the `outative` property of summations, but note that it does not remove this property, it only bypasses it.

  In some cases, a `scanmap (multthru, expr)` may be necessary before the `intosum`.

  Categories: Expressions ·

Option variable: simpproduct ¶

  Default value: `false`

  When `simpproduct` is `true`, the result of a `product` is simplified. This simplification may sometimes be able to produce a closed form. If `simpproduct` is `false` or if the quoted form `'product` is used, the value is a product noun form which is a representation of the pi notation used in mathematics.

  Categories: Sums and products · Simplification flags and variables ·

Function: product (expr, i, i_0, i_1) ¶

  Represents a product of the values of expr as the index i varies from i_0 to i_1. The noun form `'product` is displayed as an uppercase letter pi.

  `product` evaluates expr and lower and upper limits i_0 and i_1, `product` quotes (does not evaluate) the index i.

  If the upper and lower limits differ by an integer, expr is evaluated for each value of the index i, and the result is an explicit product.

  Otherwise, the range of the index is indefinite. Some rules are applied to simplify the product. When the global variable `simpproduct` is `true`, additional rules are applied. In some cases, simplification yields a result which is not a product; otherwise, the result is a noun form `'product`.

  See also `nouns` and `evflag`.

  Examples:

      (%i1) product (x + i*(i+1)/2, i, 1, 4);
      (%o1)           (x + 1) (x + 3) (x + 6) (x + 10)
      (%i2) product (i^2, i, 1, 7);
      (%o2)                       25401600
      (%i3) product (a[i], i, 1, 7);
      (%o3)                 a  a  a  a  a  a  a
                             1  2  3  4  5  6  7
      (%i4) product (a(i), i, 1, 7);
      (%o4)          a(1) a(2) a(3) a(4) a(5) a(6) a(7)
      (%i5) product (a(i), i, 1, n);
                                   n
                                 /===\
                                  ! !
      (%o5)                       ! !  a(i)
                                  ! !
                                 i = 1
      (%i6) product (k, k, 1, n);
                                     n
                                   /===\
                                    ! !
      (%o6)                         ! !  k
                                    ! !
                                   k = 1
      (%i7) product (k, k, 1, n), simpproduct;
      (%o7)                          n!
      (%i8) product (integrate (x^k, x, 0, 1), k, 1, n);
                                   n
                                 /===\
                                  ! !    1
      (%o8)                       ! !  -----
                                  ! !  k + 1
                                 k = 1
      (%i9) product (if k <= 5 then a^k else b^k, k, 1, 10);
                                    15  40
      (%o9)                        a   b

  Categories: Sums and products ·

Option variable: simpsum ¶

  Default value: `false`

  When `simpsum` is `true`, the result of a `sum` is simplified. This simplification may sometimes be able to produce a closed form. If `simpsum` is `false` or if the quoted form `'sum` is used, the value is a sum noun form which is a representation of the sigma notation used in mathematics.

  Categories: Sums and products · Simplification flags and variables ·

Function: sum (expr, i, i_0, i_1) ¶

  Represents a summation of the values of expr as the index i varies from i_0 to i_1. The noun form `'sum` is displayed as an uppercase letter sigma.

  `sum` evaluates its summand expr and lower and upper limits i_0 and i_1, `sum` quotes (does not evaluate) the index i.

  If the upper and lower limits differ by an integer, the summand expr is evaluated for each value of the summation index i, and the result is an explicit sum.

  Otherwise, the range of the index is indefinite. Some rules are applied to simplify the summation. When the global variable `simpsum` is `true`, additional rules are applied. In some cases, simplification yields a result which is not a summation; otherwise, the result is a noun form `'sum`.

  When the `evflag` (evaluation flag) `cauchysum` is `true`, a product of summations is expressed as a Cauchy product, in which the index of the inner summation is a function of the index of the outer one, rather than varying independently.

  The global variable `genindex` is the alphabetic prefix used to generate the next index of summation, when an automatically generated index is needed.

  `gensumnum` is the numeric suffix used to generate the next index of summation, when an automatically generated index is needed. When `gensumnum` is `false`, an automatically-generated index is only `genindex` with no numeric suffix.

  Note that `sum` is slow for symbolic sums that result in many terms, such as `sum(x^i, i, 1, 10000)`. For such cases, using `tree_reduce` and `makelist` is significantly faster, e.g. `tree_reduce("+", makelist(x^i, i, 1, 10000))`.

  See also `lsum`, `sumcontract`, `intosum`, `bashindices`, `niceindices`, `nouns`, `evflag`, and Package zeilberger

  Examples:

      (%i1) sum (i^2, i, 1, 7);
      (%o1)                          140
      (%i2) sum (a[i], i, 1, 7);
      (%o2)           a  + a  + a  + a  + a  + a  + a
                       7    6    5    4    3    2    1
      (%i3) sum (a(i), i, 1, 7);
      (%o3)    a(7) + a(6) + a(5) + a(4) + a(3) + a(2) + a(1)
      (%i4) sum (a(i), i, 1, n);
                                  n
                                 ====
                                 \
      (%o4)                       >    a(i)
                                 /
                                 ====
                                 i = 1
      (%i5) sum (2^i + i^2, i, 0, n);
                                n
                               ====
                               \       i    2
      (%o5)                     >    (2  + i )
                               /
                               ====
                               i = 0
      (%i6) sum (2^i + i^2, i, 0, n), simpsum;
                                    3      2
                         n + 1   2 n  + 3 n  + n
      (%o6)             2      + --------------- - 1
                                        6
      (%i7) sum (1/3^i, i, 1, inf);
                                  inf
                                  ====
                                  \     1
      (%o7)                        >    --
                                  /      i
                                  ====  3
                                  i = 1
      (%i8) sum (1/3^i, i, 1, inf), simpsum;
                                      1
      (%o8)                           -
                                      2
      (%i9) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf);
                                    inf
                                    ====
                                    \     1
      (%o9)                      30  >    --
                                    /      2
                                    ====  i
                                    i = 1
      (%i10) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf), simpsum;
                                        2
      (%o10)                       5 %pi
      (%i11) sum (integrate (x^k, x, 0, 1), k, 1, n);
                                  n
                                 ====
                                 \       1
      (%o11)                      >    -----
                                 /     k + 1
                                 ====
                                 k = 1
      (%i12) sum (if k <= 5 then a^k else b^k, k, 1, 10);
                10    9    8    7    6    5    4    3    2
      (%o12)   b   + b  + b  + b  + b  + a  + a  + a  + a  + a

  Categories: Sums and products ·

Function: sumcontract (expr) ¶

  Combines all sums of an addition that have upper and lower bounds that differ by constants. The result is an expression containing one summation for each set of such summations added to all appropriate extra terms that had to be extracted to form this sum. `sumcontract` combines all compatible sums and uses one of the indices from one of the sums if it can, and then try to form a reasonable index if it cannot use any supplied.

  It may be necessary to do an `intosum (expr)` before the `sumcontract`.

  Categories: Sums and products ·

Option variable: sumexpand ¶

  Default value: `false`

  When `sumexpand` is `true`, products of sums and exponentiated sums simplify to nested sums.

  See also `cauchysum`.

  Examples:

      (%i1) sumexpand: true$
      (%i2) sum (f (i), i, 0, m) * sum (g (j), j, 0, n);

                           m      n
                          ====   ====
                          \      \
      (%o2)                >      >     f(i1) g(i2)
                          /      /
                          ====   ====
                          i1 = 0 i2 = 0

      (%i3) sum (f (i), i, 0, m)^2;
                           m      m
                          ====   ====
                          \      \
      (%o3)                >      >     f(i3) f(i4)
                          /      /
                          ====   ====
                          i3 = 0 i4 = 0

  Categories: Sums and products · Simplification flags and variables ·

### 28.2 Introduction to Series ¶

Maxima contains functions `taylor` and `powerseries` for finding the series of differentiable functions. It also has tools such as `nusum` capable of finding the closed form of some series. Operations such as addition and multiplication work as usual on series. This section presents the global variables which control the expansion.

### 28.3 Functions and Variables for Series ¶

Option variable: cauchysum ¶

  Default value: `false`

  When multiplying together sums with `inf` as their upper limit, if `sumexpand` is `true` and `cauchysum` is `true` then the Cauchy product will be used rather than the usual product. In the Cauchy product the index of the inner summation is a function of the index of the outer one rather than varying independently.

  Example:

      (%i1) sumexpand: false$
      (%i2) cauchysum: false$

      (%i3) s: sum (f(i), i, 0, inf) * sum (g(j), j, 0, inf);
                            inf         inf
                            ====        ====
                            \           \
      (%o3)                ( >    f(i))  >    g(j)
                            /           /
                            ====        ====
                            i = 0       j = 0

      (%i4) sumexpand: true$
      (%i5) cauchysum: true$

      (%i6) expand(s,0,0);
                       inf     i1
                       ====   ====
                       \      \
      (%o6)             >      >     g(i1 - i2) f(i2)
                       /      /
                       ====   ====
                       i1 = 0 i2 = 0

  Categories: Sums and products ·

Function: deftaylor (f_1(x_1), expr_1, …, f_n(x_n), expr_n) ¶

  For each function f_i of one variable x_i, `deftaylor` defines expr_i as the Taylor series about zero. expr_i is typically a polynomial in x_i or a summation; more general expressions are accepted by `deftaylor` without complaint.

  `powerseries (f_i(x_i), x_i, 0)` returns the series defined by `deftaylor`.

  `deftaylor` returns a list of the functions f_1, …, f_n. `deftaylor` evaluates its arguments.

  Example:

      (%i1) deftaylor (f(x), x^2 + sum(x^i/(2^i*i!^2), i, 4, inf));
      (%o1)                          [f]
      (%i2) powerseries (f(x), x, 0);
                            inf
                            ====      i1
                            \        x         2
      (%o2)                  >     -------- + x
                            /       i1    2
                            ====   2   i1!
                            i1 = 4
      (%i3) taylor (exp (sqrt (f(x))), x, 0, 4);
                            2         3          4
                           x    3073 x    12817 x
      (%o3)/T/     1 + x + -- + ------- + -------- + . . .
                           2     18432     307200

  Categories: Power series ·

Option variable: maxtayorder ¶

  Default value: `true`

  When `maxtayorder` is `true`, then during algebraic manipulation of (truncated) Taylor series, `taylor` tries to retain as many terms as are known to be correct.

  Categories: Power series ·

Function: niceindices (expr) ¶

  Renames the indices of sums and products in expr. `niceindices` attempts to rename each index to the value of `niceindicespref[1]`, unless that name appears in the summand or multiplicand, in which case `niceindices` tries the succeeding elements of `niceindicespref` in turn, until an unused variable is found. If the entire list is exhausted, additional indices are constructed by appending integers to the value of `niceindicespref[1]`, e.g., `i0`, `i1`, `i2`, …

  `niceindices` returns an expression. `niceindices` evaluates its argument.

  Example:

      (%i1) niceindicespref;
      (%o1)                  [i, j, k, l, m, n]
      (%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
                       inf    inf
                      /===\   ====
                       ! !    \
      (%o2)            ! !     >      f(bar i j + foo)
                       ! !    /
                      bar = 1 ====
                              foo = 1
      (%i3) niceindices (%);

                           inf  inf
                          /===\ ====
                           ! !  \
      (%o3)                ! !   >    f(i j l + k)
                           ! !  /
                          l = 1 ====
                                k = 1

  Categories: Sums and products ·

Option variable: niceindicespref ¶

  Default value: `[i, j, k, l, m, n]`

  `niceindicespref` is the list from which `niceindices` takes the names of indices for sums and products.

  The elements of `niceindicespref` are must be names of simple variables.

  Example:

      (%i1) niceindicespref: [p, q, r, s, t, u]$
      (%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
                       inf    inf
                      /===\   ====
                       ! !    \
      (%o2)            ! !     >      f(bar i j + foo)
                       ! !    /
                      bar = 1 ====
                              foo = 1
      (%i3) niceindices (%);
                           inf  inf
                          /===\ ====
                           ! !  \
      (%o3)                ! !   >    f(i j q + p)
                           ! !  /
                          q = 1 ====
                                p = 1

  Categories: Sums and products ·

Function: nusum (expr, x, i_0, i_1) ¶

  Carries out indefinite hypergeometric summation of expr with respect to x using a decision procedure due to R.W. Gosper. expr and the result must be expressible as products of integer powers, factorials, binomials, and rational functions.

  The terms "definite" and "indefinite summation" are used analogously to "definite" and "indefinite integration". To sum indefinitely means to give a symbolic result for the sum over intervals of variable length, not just e.g. 0 to inf. Thus, since there is no formula for the general partial sum of the binomial series, `nusum` can’t do it.

  `nusum` and `unsum` know a little about sums and differences of finite products. See also `unsum`.

  Examples:

      (%i1) nusum (n*n!, n, 0, n);

      Dependent equations eliminated:  (1)
      (%o1)                     (n + 1)! - 1
      (%i2) nusum (n^4*4^n/binomial(2*n,n), n, 0, n);
                           4        3       2              n
            2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4      2
      (%o2) ------------------------------------------------ - ------
                          693 binomial(2 n, n)                 3 11 7
      (%i3) unsum (%, n);
                                    4  n
                                   n  4
      (%o3)                   ----------------
                              binomial(2 n, n)
      (%i4) unsum (prod (i^2, i, 1, n), n);
                          n - 1
                          /===\
                           ! !   2
      (%o4)              ( ! !  i ) (n - 1) (n + 1)
                           ! !
                          i = 1
      (%i5) nusum (%, n, 1, n);

      Dependent equations eliminated:  (2 3)
                                  n
                                /===\
                                 ! !   2
      (%o5)                      ! !  i  - 1
                                 ! !
                                i = 1

  Categories: Sums and products ·

Function: pade (taylor_series, numer_deg_bound, denom_deg_bound) ¶

  Returns a list of all rational functions which have the given Taylor series expansion where the sum of the degrees of the numerator and the denominator is less than or equal to the truncation level of the power series, i.e. are "best" approximants, and which additionally satisfy the specified degree bounds.

  taylor_series is an univariate Taylor series. numer_deg_bound and denom_deg_bound are positive integers specifying degree bounds on the numerator and denominator.

  taylor_series can also be a Laurent series, and the degree bounds can be `inf` which causes all rational functions whose total degree is less than or equal to the length of the power series to be returned. Total degree is defined as `numer_deg_bound + denom_deg_bound`. Length of a power series is defined as `"truncation level" + 1 - min(0, "order of series")`.

      (%i1) taylor (1 + x + x^2 + x^3, x, 0, 3);
                                    2    3
      (%o1)/T/             1 + x + x  + x  + . . .
      (%i2) pade (%, 1, 1);
                                       1
      (%o2)                       [- -----]
                                     x - 1
      (%i3) t: taylor(-(83787*x^10 - 45552*x^9 - 187296*x^8
                         + 387072*x^7 + 86016*x^6 - 1507328*x^5
                         + 1966080*x^4 + 4194304*x^3 - 25165824*x^2
                         + 67108864*x - 134217728)
             /134217728, x, 0, 10);
                          2    3       4       5       6        7
                   x   3 x    x    15 x    23 x    21 x    189 x
      (%o3)/T/ 1 - - + ---- - -- - ----- + ----- - ----- - ------
                   2    16    32   1024    2048    32768   65536

                                        8         9          10
                                  5853 x    2847 x    83787 x
                                + ------- + ------- - --------- + . . .
                                  4194304   8388608   134217728
      (%i4) pade (t, 4, 4);
      (%o4)                          []

  There is no rational function of degree 4 numerator/denominator, with this power series expansion. You must in general have degree of the numerator and degree of the denominator adding up to at least the degree of the power series, in order to have enough unknown coefficients to solve.

      (%i5) pade (t, 5, 5);
                           5                4                 3
      (%o5) [- (520256329 x  - 96719020632 x  - 489651410240 x

                        2
       - 1619100813312 x  - 2176885157888 x - 2386516803584)

                     5                 4                  3
      /(47041365435 x  + 381702613848 x  + 1360678489152 x

                        2
       + 2856700692480 x  + 3370143559680 x + 2386516803584)]

  Categories: Power series ·

Function: powerseries (expr, x, a) ¶

  Returns the general form of the power series expansion for expr in the variable x about the point a (which may be `inf` for infinity):

                 inf
                 ====
                 \               n
                  >    b  (x - a)
                 /      n
                 ====
                 n = 0

  If `powerseries` is unable to expand expr, `taylor` may give the first several terms of the series.

  When `verbose` is `true`, `powerseries` prints progress messages.

      (%i1) verbose: true$
      (%i2) powerseries (log(sin(x)/x), x, 0);
      can't expand
                                       log(sin(x))
      so we'll try again after applying the rule:
                                              d
                                            / -- (sin(x))
                                            [ dx
                              log(sin(x)) = i ----------- dx
                                            ]   sin(x)
                                            /
      in the first simplification we have returned:
                                   /
                                   [
                                   i cot(x) dx - log(x)
                                   ]
                                   /
                          inf
                          ====        i1  2 i1             2 i1
                          \      (- 1)   2     bern(2 i1) x
                           >     ------------------------------
                          /                i1 (2 i1)!
                          ====
                          i1 = 1
      (%o2)                -------------------------------------
                                            2

  Categories: Power series ·

Option variable: psexpand ¶

  Default value: `false`

  When `psexpand` is `true`, an extended rational function expression is displayed fully expanded. The switch `ratexpand` has the same effect.

  When `psexpand` is `false`, a multivariate expression is displayed just as in the rational function package.

  When `psexpand` is `multi`, then terms with the same total degree in the variables are grouped together.

  Categories: Display flags and variables ·

Function: revert (expr, x) ¶
Function: revert2 (expr, x, n) ¶

  These functions return the reversion of expr, a Taylor series about zero in the variable x. `revert` returns a polynomial of degree equal to the highest power in expr. `revert2` returns a polynomial of degree n, which may be greater than, equal to, or less than the degree of expr.

  `load ("revert")` loads these functions.

  Examples:

      (%i1) load ("revert")$
      (%i2) t: taylor (exp(x) - 1, x, 0, 6);
                         2    3    4    5     6
                        x    x    x    x     x
      (%o2)/T/      x + -- + -- + -- + --- + --- + . . .
                        2    6    24   120   720
      (%i3) revert (t, x);
                     6       5       4       3       2
                 10 x  - 12 x  + 15 x  - 20 x  + 30 x  - 60 x
      (%o3)/R/ - --------------------------------------------
                                      60
      (%i4) ratexpand (%);
                           6    5    4    3    2
                          x    x    x    x    x
      (%o4)             - -- + -- - -- + -- - -- + x
                          6    5    4    3    2
      (%i5) taylor (log(x+1), x, 0, 6);
                          2    3    4    5    6
                         x    x    x    x    x
      (%o5)/T/       x - -- + -- - -- + -- - -- + . . .
                         2    3    4    5    6
      (%i6) ratsimp (revert (t, x) - taylor (log(x+1), x, 0, 6));
      (%o6)                           0
      (%i7) revert2 (t, x, 4);
                                4    3    2
                               x    x    x
      (%o7)                  - -- + -- - -- + x
                               4    3    2

  Categories: Power series ·

Function: taylor
taylor (expr, x, a, n)
taylor (expr, [x_1, x_2, …], a, n)
taylor (expr, [x, a, n, ’asymp])
taylor (expr, [x_1, x_2, …], [a_1, a_2, …], [n_1, n_2, …])
taylor (expr, [x_1, a_1, n_1], [x_2, a_2, n_2], …) ¶

  `taylor (expr, x, a, n)` expands the expression expr in a truncated Taylor or Laurent series in the variable x around the point a, containing terms through `(x - a)^n`.

  If expr is of the form `f(x)/g(x)` and `g(x)` has no terms up to degree n then `taylor` attempts to expand `g(x)` up to degree `2 n`. If there are still no nonzero terms, `taylor` doubles the degree of the expansion of `g(x)` so long as the degree of the expansion is less than or equal to `n 2^taylordepth`.

  `taylor (expr, [x_1, x_2, ...], a, n)` returns a truncated power series of degree n in all variables x_1, x_2, … about the point `(a, a, ...)`.

  `taylor (expr, [x_1, a_1, n_1], [x_2, a_2, n_2], ...)` returns a truncated power series in the variables x_1, x_2, … about the point `(a_1, a_2, ...)`, truncated at n_1, n_2, …

  `taylor (expr, [x_1, x_2, ...], [a_1, a_2, ...], [n_1, n_2, ...])` returns a truncated power series in the variables x_1, x_2, … about the point `(a_1, a_2, ...)`, truncated at n_1, n_2, …

  `taylor (expr, [x, a, n, 'asymp])` returns an expansion of expr in negative powers of `x - a`. The highest order term is `(x - a)^-n`.

  When `maxtayorder` is `true`, then during algebraic manipulation of (truncated) Taylor series, `taylor` tries to retain as many terms as are known to be correct.

  When `psexpand` is `true`, an extended rational function expression is displayed fully expanded. The switch `ratexpand` has the same effect. When `psexpand` is `false`, a multivariate expression is displayed just as in the rational function package. When `psexpand` is `multi`, then terms with the same total degree in the variables are grouped together.

  See also the `taylor_logexpand` switch for controlling expansion.

  Examples:

      (%i1) taylor (sqrt (sin(x) + a*x + 1), x, 0, 3);
                                 2             2
                   (a + 1) x   (a  + 2 a + 1) x
      (%o1)/T/ 1 + --------- - -----------------
                       2               8

                                         3      2             3
                                     (3 a  + 9 a  + 9 a - 1) x
                                   + -------------------------- + . . .
                                                 48
      (%i2) %^2;
                                          3
                                         x
      (%o2)/T/           1 + (a + 1) x - -- + . . .
                                         6
      (%i3) taylor (sqrt (x + 1), x, 0, 5);
                             2    3      4      5
                        x   x    x    5 x    7 x
      (%o3)/T/      1 + - - -- + -- - ---- + ---- + . . .
                        2   8    16   128    256
      (%i4) %^2;
      (%o4)/T/                  1 + x + . . .
      (%i5) product ((1 + x^i)^2.5, i, 1, inf)/(1 + x^2);

                               inf
                              /===\
                               ! !    i     2.5
                               ! !  (x  + 1)
                               ! !
                              i = 1
      (%o5)                   -----------------
                                    2
                                   x  + 1

      (%i6) ev (taylor(%, x,  0, 3), keepfloat);
                                     2           3
      (%o6)/T/    1 + 2.5 x + 3.375 x  + 6.5625 x  + . . .
      (%i7) taylor (1/log (x + 1), x, 0, 3);
                                     2       3
                       1   1   x    x    19 x
      (%o7)/T/         - + - - -- + -- - ----- + . . .
                       x   2   12   24    720
      (%i8) taylor (cos(x) - sec(x), x, 0, 5);
                                      4
                                 2   x
      (%o8)/T/                - x  - -- + . . .
                                     6
      (%i9) taylor ((cos(x) - sec(x))^3, x, 0, 5);
      (%o9)/T/                    0 + . . .
      (%i10) taylor (1/(cos(x) - sec(x))^3, x, 0, 5);
                                                     2          4
                  1     1       11      347    6767 x    15377 x
      (%o10)/T/ - -- + ---- + ------ - ----- - ------- - --------
                   6      4        2   15120   604800    7983360
                  x    2 x    120 x

                                                                + . . .
      (%i11) taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6);
                     2  2       4      2   4
                    k  x    (3 k  - 4 k ) x
      (%o11)/T/ 1 - ----- - ----------------
                      2            24

                                          6       4       2   6
                                     (45 k  - 60 k  + 16 k ) x
                                   - -------------------------- + . . .
                                                720
      (%i12) taylor ((x + 1)^n, x, 0, 4);

                            2       2     3      2         3
                          (n  - n) x    (n  - 3 n  + 2 n) x
      (%o12)/T/ 1 + n x + ----------- + --------------------
                               2                 6

                                     4      3       2         4
                                   (n  - 6 n  + 11 n  - 6 n) x
                                 + ---------------------------- + . . .
                                                24

      (%i13) taylor (sin (y + x), x, 0, 3, y, 0, 3);
                     3                 2
                    y                 y
      (%o13)/T/ y - -- + . . . + (1 - -- + . . .) x
                    6                 2

                          3                       2
                     y   y            2      1   y            3
                + (- - + -- + . . .) x  + (- - + -- + . . .) x  + . . .
                     2   12                  6   12
      (%i14) taylor (sin (y + x), [x, y], 0, 3);
                           3        2      2      3
                          x  + 3 y x  + 3 y  x + y
      (%o14)/T/   y + x - ------------------------- + . . .
                                      6
      (%i15) taylor (1/sin (y + x), x, 0, 3, y, 0, 3);
                1   y              1    1               1            2
      (%o15)/T/ - + - + . . . + (- -- + - + . . .) x + (-- + . . .) x
                y   6               2   6                3
                                   y                    y

                                                 1            3
                                            + (- -- + . . .) x  + . . .
                                                  4
                                                 y
      (%i16) taylor (1/sin (y + x), [x, y], 0, 3);
                                   3         2       2        3
                  1     x + y   7 x  + 21 y x  + 21 y  x + 7 y
      (%o16)/T/ ----- + ----- + ------------------------------- + . . .
                x + y     6                   360

  Categories: Power series ·

Option variable: taylordepth ¶

  Default value: 3

  If there are still no nonzero terms, `taylor` doubles the degree of the expansion of `g(x)` so long as the degree of the expansion is less than or equal to `n 2^taylordepth`.

  Categories: Power series ·

Function: taylorinfo (expr) ¶

  Returns information about the Taylor series expr. The return value is a list of lists. Each list comprises the name of a variable, the point of expansion, and the degree of the expansion.

  `taylorinfo` returns `false` if expr is not a Taylor series.

  Example:

      (%i1) taylor ((1 - y^2)/(1 - x), x, 0, 3, [y, a, inf]);
                        2                       2
      (%o1)/T/ - (y - a)  - 2 a (y - a) + (1 - a )

               2                        2
       + (1 - a  - 2 a (y - a) - (y - a) ) x

               2                        2   2
       + (1 - a  - 2 a (y - a) - (y - a) ) x

               2                        2   3
       + (1 - a  - 2 a (y - a) - (y - a) ) x  + . . .
      (%i2) taylorinfo(%);
      (%o2)               [[y, a, inf], [x, 0, 3]]

  Categories: Power series ·

Function: taylorp (expr) ¶

  Returns `true` if expr is a Taylor series, and `false` otherwise.

  Categories: Predicate functions · Power series ·

Option variable: taylor_logexpand ¶

  Default value: `true`

  `taylor_logexpand` controls expansions of logarithms in `taylor` series.

  When `taylor_logexpand` is `true`, all logarithms are expanded fully so that zero-recognition problems involving logarithmic identities do not disturb the expansion process. However, this scheme is not always mathematically correct since it ignores branch information.

  When `taylor_logexpand` is set to `false`, then the only expansion of logarithms that occur is that necessary to obtain a formal power series.

  Categories: Power series · Exponential and logarithm functions ·

Option variable: taylor_order_coefficients ¶

  Default value: `true`

  `taylor_order_coefficients` controls the ordering of coefficients in a Taylor series.

  When `taylor_order_coefficients` is `true`, coefficients of taylor series are ordered canonically.

  Categories: Power series ·

Function: taylor_simplifier (expr) ¶

  Simplifies coefficients of the power series expr. `taylor` calls this function.

  Categories: Power series ·

Option variable: taylor_truncate_polynomials ¶

  Default value: `true`

  When `taylor_truncate_polynomials` is `true`, polynomials are truncated based upon the input truncation levels.

  Otherwise, polynomials input to `taylor` are considered to have infinite precision.

  Categories: Power series ·

Function: taytorat (expr) ¶

  Converts expr from `taylor` form to canonical rational expression (CRE) form. The effect is the same as `rat (ratdisrep (expr))`, but faster.

  Categories: Power series · Rational expressions ·

Function: trunc (expr) ¶

  Annotates the internal representation of the general expression expr so that it is displayed as if its sums were truncated Taylor series. expr is not otherwise modified.

  Example:

      (%i1) expr: x^2 + x + 1;
                                  2
      (%o1)                      x  + x + 1
      (%i2) trunc (expr);
                                      2
      (%o2)                  1 + x + x  + . . .
      (%i3) is (expr = trunc (expr));
      (%o3)                         true

  Categories: Power series ·

Function: unsum (f, n) ¶

  Returns the first backward difference `f(n) - f(n - 1)`. Thus `unsum` in a sense is the inverse of `sum`.

  See also `nusum`.

  Examples:

      (%i1) g(p) := p*4^n/binomial(2*n,n);
                                           n
                                        p 4
      (%o1)               g(p) := ----------------
                                  binomial(2 n, n)
      (%i2) g(n^4);
                                    4  n
                                   n  4
      (%o2)                   ----------------
                              binomial(2 n, n)
      (%i3) nusum (%, n, 0, n);
                           4        3       2              n
            2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4      2
      (%o3) ------------------------------------------------ - ------
                          693 binomial(2 n, n)                 3 11 7
      (%i4) unsum (%, n);
                                    4  n
                                   n  4
      (%o4)                   ----------------
                              binomial(2 n, n)

  Categories: Sums and products ·

Option variable: verbose ¶

  Default value: `false`

  When `verbose` is `true`, `powerseries` prints progress messages.

  Categories: Power series ·

### 28.4 Introduction to Fourier series ¶

The `fourie` package comprises functions for the symbolic computation of Fourier series. There are functions in the `fourie` package to calculate Fourier integral coefficients and some functions for manipulation of expressions.

Categories: Fourier transform · Share packages · Package fourie ·

### 28.5 Functions and Variables for Fourier series ¶

Function: equalp (x, y) ¶

  Returns `true` if `equal (x, y)` otherwise `false` (doesn’t give an error message like `equal (x, y)` would do in this case).

  Categories: Package fourie ·

Function: remfun
remfun (f, expr)
remfun (f, expr, x) ¶

  `remfun (f, expr)` replaces all occurrences of `f (arg)` by arg in expr.

  `remfun (f, expr, x)` replaces all occurrences of `f (arg)` by arg in expr only if arg contains the variable x.

  Categories: Package fourie ·

Function: funp
funp (f, expr)
funp (f, expr, x) ¶

  `funp (f, expr)` returns `true` if expr contains the function f.

  `funp (f, expr, x)` returns `true` if expr contains the function f and the variable x is somewhere in the argument of one of the instances of f.

  Categories: Package fourie ·

Function: absint
absint (f, x, halfplane)
absint (f, x)
absint (f, x, a, b) ¶

  `absint (f, x, halfplane)` returns the indefinite integral of f with respect to x in the given halfplane (`pos`, `neg`, or `both`). f may contain expressions of the form `abs (x)`, `abs (sin (x))`, `abs (a) * exp (-abs (b) * abs (x))`.

  `absint (f, x)` is equivalent to `absint (f, x, pos)`.

  `absint (f, x, a, b)` returns the definite integral of f with respect to x from a to b. f may include absolute values.

  Categories: Package fourie · Integral calculus ·

Function: fourier (f, x, p) ¶

  Returns a list of the Fourier coefficients of `f(x)` defined on the interval `[-p, p]`.

  Categories: Package fourie ·

Function: foursimp (l) ¶

  Simplifies `sin (n %pi)` to 0 if `sinnpiflag` is `true` and `cos (n %pi)` to `(-1)^n` if `cosnpiflag` is `true`.

  Categories: Package fourie · Trigonometric functions · Simplification functions ·

Option variable: sinnpiflag ¶

  Default value: `true`

  See `foursimp`.

  Categories: Package fourie ·

Option variable: cosnpiflag ¶

  Default value: `true`

  See `foursimp`.

  Categories: Package fourie ·

Function: fourexpand (l, x, p, limit) ¶

  Constructs and returns the Fourier series from the list of Fourier coefficients l up through limit terms (limit may be `inf`). x and p have same meaning as in `fourier`.

  Categories: Package fourie ·

Function: fourcos (f, x, p) ¶

  Returns the Fourier cosine coefficients for `f(x)` defined on `[0, p]`.

  Categories: Package fourie ·

Function: foursin (f, x, p) ¶

  Returns the Fourier sine coefficients for `f(x)` defined on `[0, p]`.

  Categories: Package fourie ·

Function: totalfourier (f, x, p) ¶

  Returns `fourexpand (foursimp (fourier (f, x, p)), x, p, 'inf)`.

  Categories: Package fourie ·

Function: fourint (f, x) ¶

  Constructs and returns a list of the Fourier integral coefficients of `f(x)` defined on `[minf, inf]`.

  Categories: Package fourie ·

Function: fourintcos (f, x) ¶

  Returns the Fourier cosine integral coefficients for `f(x)` on `[0, inf]`.

  Categories: Package fourie ·

Function: fourintsin (f, x) ¶

  Returns the Fourier sine integral coefficients for `f(x)` on `[0, inf]`.

  Categories: Package fourie ·

### 28.6 Functions and Variables for Poisson series ¶

Function: intopois (a) ¶

  Converts a into a Poisson encoding.

  Categories: Poisson series ·

Function: outofpois (a) ¶

  Converts a from Poisson encoding to general representation. If a is not in Poisson form, `outofpois` carries out the conversion, i.e., the return value is `outofpois (intopois (a))`. This function is thus a canonical simplifier for sums of powers of sine and cosine terms of a particular type.

  Categories: Poisson series ·

Function: poisdiff (a, b) ¶

  Differentiates a with respect to b. b must occur only in the trig arguments or only in the coefficients.

  Categories: Poisson series ·

Function: poisexpt (a, b) ¶

  Functionally identical to `intopois (a^b)`. b must be a positive integer.

  Categories: Poisson series ·

Function: poisint (a, b) ¶

  Integrates in a similarly restricted sense (to `poisdiff`). Non-periodic terms in b are dropped if b is in the trig arguments.

  Categories: Poisson series ·

Option variable: poislim ¶

  Default value: 5

  `poislim` determines the domain of the coefficients in the arguments of the trig functions. The initial value of 5 corresponds to the interval [-2^(5-1)+1,2^(5-1)], or [-15,16], but it can be set to [-2^(n-1)+1, 2^(n-1)].

  Categories: Poisson series ·

Function: poismap (series, sinfn, cosfn) ¶

  will map the functions sinfn on the sine terms and cosfn on the cosine terms of the Poisson series given. sinfn and cosfn are functions of two arguments which are a coefficient and a trigonometric part of a term in series respectively.

  Categories: Poisson series ·

Function: poisplus (a, b) ¶

  Is functionally identical to `intopois (a + b)`.

  Categories: Poisson series ·

Function: poissimp (a) ¶

  Converts a into a Poisson series for a in general representation.

  Categories: Poisson series ·

Special symbol: poisson ¶

  The symbol `/P/` follows the line label of Poisson series expressions.

  Categories: Poisson series ·

Function: poissubst (a, b, c) ¶

  Substitutes a for b in c. c is a Poisson series.

  (1) Where B is a variable u, v, w, x, y, or z, then a must be an expression linear in those variables (e.g., `6*u + 4*v`).

  (2) Where b is other than those variables, then a must also be free of those variables, and furthermore, free of sines or cosines.

  `poissubst (a, b, c, d, n)` is a special type of substitution which operates on a and b as in type (1) above, but where d is a Poisson series, expands `cos(d)` and `sin(d)` to order n so as to provide the result of substituting `a + d` for b in c. The idea is that d is an expansion in terms of a small parameter. For example, `poissubst (u, v, cos(v), %e, 3)` yields `cos(u)*(1 - %e^2/2) - sin(u)*(%e - %e^3/6)`.

  Categories: Poisson series ·

Function: poistimes (a, b) ¶

  Is functionally identical to `intopois (a*b)`.

  Categories: Poisson series ·

Function: poistrim () ¶

  is a reserved function name which (if the user has defined it) gets applied during Poisson multiplication. It is a predicate function of 6 arguments which are the coefficients of the u, v, ..., z in a term. Terms for which `poistrim` is `true` (for the coefficients of that term) are eliminated during multiplication.

  Categories: Poisson series ·

Function: printpois (a) ¶

  Prints a Poisson series in a readable format. In common with `outofpois`, it will convert a into a Poisson encoding first, if necessary.

  Categories: Poisson series · Display functions ·