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28.1 Sums and Products | ||

28.2 Introduction to Series | ||

28.3 Functions and Variables for Series | ||

28.4 Poisson series |

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__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`j`

. The quantity`number``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)); ==== \ 2 (%o2) > i / ==== 3 i in rootsof(x - 1)

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,`

may be necessary before the`expr`)`intosum`

.Categories: Expressions

__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.See also

`sumcontract`

,`intosum`

,`bashindices`

,`niceindices`

,`nouns`

,`evflag`

, and`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 (`

before the`expr`)`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

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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.

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__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) ''s; 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 (`

returns the series defined by`f_i`(`x_i`),`x_i`, 0)`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 typically names of variables, although that is not enforced by`niceindices`

.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 a 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

. Length of a power series is defined as`numer_deg_bound`+`denom_deg_bound``"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***(*`expr`,`x`,`a`,`n`)__Function:__**taylor***(*`expr`, [`x_1`,`x_2`, ...],`a`,`n`)__Function:__**taylor***(*`expr`, [`x`,`a`,`n`, 'asymp])__Function:__**taylor***(*`expr`, [`x_1`,`x_2`, …], [`a_1`,`a_2`, …], [`n_1`,`n_2`, …])__Function:__**taylor***(*`expr`, [`x_1`,`a_1`,`n_1`], [`x_2`,`a_2`,`n_2`], …)`taylor (`

expands the expression`expr`,`x`,`a`,`n`)`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

and`f`(`x`)/`g`(`x`)

has no terms up to degree`g`(`x`)`n`then`taylor`

attempts to expand

up to degree`g`(`x`)`2`

. If there are still no nonzero terms,`n``taylor`

doubles the degree of the expansion of

so long as the degree of the expansion is less than or equal to`g`(`x`)

.`n`2^taylordepth`taylor (`

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

.`a`,`a`, ...)`taylor (`

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

, truncated at`a_1`,`a_2`, ...)`n_1`,`n_2`, …`taylor (`

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

, truncated at`a_1`,`a_2`, ...)`n_1`,`n_2`, …`taylor (`

returns an expansion of`expr`, [`x`,`a`,`n`, 'asymp])`expr`in negative powers of

. The highest order term is`x`-`a``(`

.`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

so long as the degree of the expansion is less than or equal to`g`(`x`)

.`n`2^taylordepthCategories: 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 precison.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 (`

, but faster.`expr`))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

. Thus`f`(`n`) -`f`(`n`- 1)`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

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__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 (`

. This function is thus a canonical simplifier for sums of powers of sine and cosine terms of a particular type.`a`))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 (`

is a special type of substitution which operates on`a`,`b`,`c`,`d`,`n`)`a`and`b`as in type (1) above, but where`d`is a Poisson series, expands`cos(`

and`d`)`sin(`

to order`d`)`n`so as to provide the result of substituting

for`a`+`d``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

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This document was generated by *Oliver Kullmann* on *May, 18 2013* using *texi2html 1.76*.