The int command now uses differential equation routines for exact ODEs to compute integrals when the integrand contains unknown functions and is a total derivative with respect to those functions.

Example

Int((3*diff(u(x),x)*v(x)^2-diff(u(x),x)^3)*sin(u(x)) + (-6*v(x)*diff(v(x),x)+2*diff(u(x),x)*diff(u(x),x,x))*cos(u(x)) + 8*diff(v(x),x)*diff(v(x),x,x), x);

$\int \left(\left(3\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right){v\left(x\right)}^2-{\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)}^3\right)\sin \left(u\left(x\right)\right)+\left(-6v\left(x\right)\left(\frac{\ⅆ}{\ⅆx}v\left(x\right)\right)+2\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\right)\right)\cos \left(u\left(x\right)\right)+8\left(\frac{\ⅆ}{\ⅆx}v\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}v\left(x\right)\right)\right)\ⅆx$

value((1));

${\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)}^2\cos \left(u\left(x\right)\right)-3{v\left(x\right)}^2\cos \left(u\left(x\right)\right)+4{\left(\frac{\ⅆ}{\ⅆx}v\left(x\right)\right)}^2$

The differential equation solving capabilities of the dsolve and pdsolve commands, for computing both exact and numeric solutions, have been enhanced with new algorithms (see Updates to Differential Equation Solvers). Also, seven new commands, some based on original algorithms, are available in DEtools and PDEtools for this release. For a description of these new commands, see Updates to Differential Equation Solvers: Packages.

The sum command has been greatly enhanced by rewriting it to use the SumTools[Summation] routine.

Examples

More closed forms by using accurate summation:

Sum(Psi(x)^2,x);

$\sum _x{\mathrm{\Psi }\left(x\right)}^2$

(3) = value((3));

$\sum _x{\mathrm{\Psi }\left(x\right)}^2=\frac{\left(4x^4-16x^3-21x^2-4x-2\right){\mathrm{\Psi }\left(x\right)}^2}{2\left(2x+1\right)}+\left(-2x^3+9x^2+13x+\frac{1}{2}\right){\mathrm{\Psi }\left(x+1\right)}^2+\frac{{\left(x+1\right)}^2\left(4x^2-24x-1\right){\mathrm{\Psi }\left(x+2\right)}^2}{2\left(2x+1\right)}$

More partial results via additive decomposition:

Sum(5^x*(3*x-2)/x/(x+1), x);

$\sum _x\frac{5^x\left(3x-2\right)}{x\left(x+1\right)}$

(5) = value((5));

$\sum _x\frac{5^x\left(3x-2\right)}{x\left(x+1\right)}=\frac{25^x}{x}+\sum _x\left(-\frac{55^x}{x+1}\right)$

Nicer results for some definite sums:

Sum(binomial(n,4*k),k=0..infinity);

$\sum _{k=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0ex}{}{n}{4k}\right)$

(7) = value((7));

$\sum _{k=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0ex}{}{n}{4k}\right)=\frac{2^n}{4}+\frac{{\left(1+\mathrm{I}\right)}^n}{4}+\frac{{\left(1-\mathrm{I}\right)}^n}{4}$

The SumTools[IndefiniteSum] package also provides an extension mechanism that is used by sum. Using this facility, you can add a summation definition related to other special functions.

The simplification of different types of expressions has been greatly enhanced in Maple 9.5.

Examples

The FunctionAdvisor command provides supporting information on mathematical properties of mathematical functions, for example, on definitions, identities, integral forms, series expansions, and others. For Maple 9.5, the FunctionAdvisor command has been enhanced regarding the computation of non-trivial "specializations" of mathematical functions for particular values of their parameters. This permits answering questions like {"In terms of which functions could a given function, evaluated at some values of its parameters, be expressed?"}

Examples

In terms of which functions could Zeta(0, z, nu) be expressed?

FunctionAdvisor(specialize, Zeta(0, z, nu));

$\left[{\mathrm{\zeta }}^{\left(0\right)}\left(z\&comma;\mathrm{\nu }\right)=\mathrm{LerchPhi}\left(1\&comma;z\&comma;\mathrm{\nu }\right)\&comma;\mathrm{with\; no\; restrictions\; on}\left(z\&comma;\mathrm{\nu }\right)\right],\left[{\mathrm{\zeta }}^{\left(0\right)}\left(z\&comma;\mathrm{\nu }\right)=\frac{{\left(\mathrm{-1}\right)}^z\mathrm{\Psi }\left(z-1\&comma;\mathrm{\nu }\right)}{\left(z-1\right)!}\&comma;z::\mathbb{Z}\wedge 1<z\right],\left[{\mathrm{\zeta }}^{\left(0\right)}\left(z\&comma;\mathrm{\nu }\right)=\mathrm{\zeta }\left(z\right)-\mathrm{harmonic}\left(\mathrm{\nu }-1\&comma;z\right)\&comma;\left(\mathrm{\nu }-1\right)::\left(\neg {\mathbb{Z}}^{-}\right)\right]$

Regarding the specialization into Psi, what is the specialization of Psi into Zeta if we ignore the restriction $z::\mathbb{Z}\wedge 1<z$ on the value of $z$?

FunctionAdvisor(specialize, Psi(z-1,nu), Zeta);

$\left[\mathrm{\Psi }\left(z-1\&comma;\mathrm{\nu }\right)=\frac{{\mathrm{\zeta }}^{\left(1\right)}\left(z\&comma;\mathrm{\nu }\right)+\left(\mathrm{\gamma }+\mathrm{\Psi }\left(1-z\right)\right){\mathrm{\zeta }}^{\left(0\right)}\left(z\&comma;\mathrm{\nu }\right)}{\mathrm{\Gamma }\left(1-z\right)}\&comma;\mathrm{with\; no\; restrictions\; on}\left(\mathrm{\nu }\&comma;z\right)\right]$

Non-trivial specializations for the hypergeometric function:

FunctionAdvisor(specialize, hypergeom([1, 1, 1 - n], [2, 2], 1) );

$\left[\mathrm{hypergeom}\left(\left[1\&comma;1\&comma;-n+1\right]\&comma;\left[2\&comma;2\right]\&comma;1\right)=\mathrm{LerchPhi}\left(1\&comma;2\&comma;1\right)\&comma;n=0\right],\left[\mathrm{hypergeom}\left(\left[1\&comma;1\&comma;-n+1\right]\&comma;\left[2\&comma;2\right]\&comma;1\right)=\frac{\mathrm{MeijerG}\left(\left[\left[0\&comma;0\&comma;n\right]\&comma;\left[\right]\right]\&comma;\left[\left[0\right]\&comma;\left[\mathrm{-1}\&comma;\mathrm{-1}\right]\right]\&comma;\mathrm{-1}\right)}{\mathrm{\Gamma }\left(-n+1\right)}\&comma;\mathrm{with\; no\; restrictions\; on}\left(n\right)\right],\left[\mathrm{hypergeom}\left(\left[1\&comma;1\&comma;-n+1\right]\&comma;\left[2\&comma;2\right]\&comma;1\right)=\frac{\mathrm{\Psi }\left(n+1\right)+\mathrm{\gamma }}{n}\&comma;\mathrm{with\; no\; restrictions\; on}\left(n\right)\right],\left[\mathrm{hypergeom}\left(\left[1\&comma;1\&comma;-n+1\right]\&comma;\left[2\&comma;2\right]\&comma;1\right)=\mathrm{dilog}\left(0\right)\&comma;n=0\right],\left[\mathrm{hypergeom}\left(\left[1\&comma;1\&comma;-n+1\right]\&comma;\left[2\&comma;2\right]\&comma;1\right)=\mathrm{polylog}\left(2\&comma;1\right)\&comma;n=0\wedge 1\ne 0\right]$

The enhanced routines are sensitive to assumptions. The specialization of $\mathrm{BesselI}\left(a\&comma;z\right)$ in terms of the $\mathrm{StruveL}$ function assuming $a+\frac{1}{2}$ is a negative integer:

FunctionAdvisor(specialize, BesselI(a,z), StruveL) assuming (a+1/2)::negint;

$\left[\mathrm{BesselI}\left(a\&comma;z\right)=-\frac{z^a\left(\frac{\mathrm{I}\mathrm{StruveL}\left(-a\&comma;-z\right){\left(-z\right)}^a}{{\left(\mathrm{I}z\right)}^{2a}}+\frac{\sum _{\mathrm{\_k1}=0}^{-\frac{1}{2}-a}\frac{{\left(\mathrm{-1}\right)}^{\mathrm{\_k1}}\mathrm{pochhammer}\left(a+\frac{1}{2}\&comma;\mathrm{\_k1}\right)\mathrm{pochhammer}\left(\frac{1}{2}\&comma;\mathrm{\_k1}\right)}{{\left(\frac{\mathrm{I}}{2}z\right)}^{a+1+2\mathrm{\_k1}}\left(-\frac{1}{2}-a\right)!\sqrt{\mathrm{\pi }}}}{{\left(\mathrm{I}z\right)}^a}\right)}{{\left(\mathrm{-1}\right)}^{a-\frac{1}{2}}}\&comma;\mathrm{with\; no\; restrictions\; on}\left(a\&comma;z\right)\right]$

To compute all the specializations for $\mathrm{BesselY}\left(\frac{1}{2}\&comma;z\right)$ that the FunctionAdvisor command can compute, enter $\mathrm{FunctionAdvisor}\left(\mathrm{specialize}\&comma;'\mathrm{BesselY}'\left(\frac{1}{2}\&comma;z\right)\right)$ at the Maple prompt.

The implementation of some mathematical functions has been improved to increase the knowledge about special values and relationships. More functions are sensitive to assumptions.

Examples

Elliptic functions:

EllipticNome( EllipticModulus(q) ) assuming abs(q) < 1;

$q$

EllipticModulus( EllipticNome(k) ) assuming Re(k) > 0;

$k$

binomial function:

binomial(a,k) assuming a::posint, k::negint;

$0$

The classic definition for the Psi(n,z) function as the nth derivative of $\mathrm{\Psi }\left(z\right)$, which requires that $n$ be a positive integer, has been extended to arbitrary complex values of $n$ in terms of the Hurwitz Zeta function and its first derivative.

FunctionAdvisor(analytic_extension, Psi(n,z));

$\mathrm{\Psi }\left(n\&comma;z\right)=\frac{{\mathrm{\zeta }}^{\left(1\right)}\left(n+1\&comma;z\right)+\left(\mathrm{\gamma }+\mathrm{\Psi }\left(-n\right)\right){\mathrm{\zeta }}^{\left(0\right)}\left(n+1\&comma;z\right)}{\mathrm{\Gamma }\left(-n\right)}$

The network of conversions between mathematical functions was introduced in Maple 8. In Maple 9, the number of conversions was almost tripled. In Maple 9.5, more conversion routines and conversion rules were added.

Examples

For functions related to special cases of the hypergeom function:

polylog(2,z);

$\mathrm{polylog}\left(2\&comma;z\right)$

(47) = convert((47), hypergeom);

$\mathrm{polylog}\left(2\&comma;z\right)=z\mathrm{hypergeom}\left(\left[1\&comma;1\&comma;1\right]\&comma;\left[2\&comma;2\right]\&comma;z\right)$

convert((48),LerchPhi);

$z\mathrm{LerchPhi}\left(z\&comma;2\&comma;1\right)=z\mathrm{LerchPhi}\left(z\&comma;2\&comma;1\right)$

convert(lhs((49)),polylog) = convert(rhs((49)),hypergeom);    # convert the other way around

$\mathrm{polylog}\left(2\&comma;z\right)=z\mathrm{hypergeom}\left(\left[1\&comma;1\&comma;1\right]\&comma;\left[2\&comma;2\right]\&comma;z\right)$

For the argument function:

argument(z);

$\arg \left(z\right)$

(51)  =  convert((51),ln);

$\arg \left(z\right)=\mathrm{-I}\ln \left(\frac{z}{\left|z\right|}\right)$

(51) = convert((51),arctan);

$\arg \left(z\right)=\arctan \left(\mathrm{\Im }\left(z\right)\&comma;\mathrm{\Re }\left(z\right)\right)$

More Sum representations for special functions:

EllipticModulus(q);

$\mathrm{EllipticModulus}\left(q\right)$

(54) = convert((54), Sum) assuming abs(q) < 1;

$\mathrm{EllipticModulus}\left(q\right)=\frac{{\left(\sum _{\mathrm{\_k1}=0}^{\mathrm{\infty }}2q^{{\left(\frac{1}{2}+\mathrm{\_k1}\right)}^2}\right)}^2}{{\left(\left(\sum _{\mathrm{\_k1}=1}^{\mathrm{\infty }}2q^{{\mathrm{\_k1}}^2}\right)+1\right)}^2}$

To reveal these sum representations without knowing in advance the required assumption on the function parameters, or the syntax of the function, use the FunctionAdvisor command.

FunctionAdvisor( sum, EllipticNome );

* Partial match of "sum" against topic "sum_form".

$\left[\mathrm{EllipticNome}\left(k\right)=\sum _{\mathrm{\_k2}=0}^{\mathrm{\infty }}\frac{{\left(-\frac{\left(\sum _{\mathrm{\_k1}=0}^{\mathrm{\infty }}\frac{{\mathrm{pochhammer}\left(\frac{1}{2}\&comma;\mathrm{\_k1}\right)}^2{\left(-k^2+1\right)}^{\mathrm{\_k1}}}{\mathrm{\_k1}!\mathrm{pochhammer}\left(1\&comma;\mathrm{\_k1}\right)}\right)\mathrm{\pi }}{\sum _{\mathrm{\_k1}=0}^{\mathrm{\infty }}\frac{{\mathrm{pochhammer}\left(\frac{1}{2}\&comma;\mathrm{\_k1}\right)}^2{k}^{2\mathrm{\_k1}}}{\mathrm{\_k1}!\mathrm{pochhammer}\left(1\&comma;\mathrm{\_k1}\right)}}\right)}^{\mathrm{\_k2}}}{\mathrm{\_k2}!}\&comma;\left|k\right|<1\right]$

New conversion rules for the hypergeometric 2F1 function

hypergeom([a,b],[c],z);

$\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;z\right)$

convert((57), hypergeom, "quadratic 1") assuming -1/2+a+b-c = 0;

$\frac{\mathrm{hypergeom}\left(\left[2a-1\&comma;-1+2b\right]\&comma;\left[-\frac{1}{2}+a+b\right]\&comma;\frac{1}{2}-\frac{\sqrt{1-z}}{2}\right)}{\sqrt{1-z}}$

The new rules available for the 2F1 function are: "quadratic n", with n between 1 and 6; "2a2b"; "raise 1/2"; and "lower 1/2". To see under which assumptions these identities are valid, use FunctionAdvisor(identities, hypergeom([a,b],[c],z)). For more information, see convert[function_rules].

Examples

More convenient and compact form for the differentiation rule of the incomplete GAMMA(a,z) function in terms of the MeijerG function:

diff(GAMMA(a,z), a);

$\mathrm{\Gamma }\left(a\&comma;z\right)\ln \left(z\right)+\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[1\&comma;1\right]\right]\&comma;\left[\left[0\&comma;0\&comma;a\right]\&comma;\left[\right]\right]\&comma;z\right)$

To express the previous expression in terms of the hypergeometric function, use $\mathrm{convert}\left(\left[\mathrm{Typesetting}:-\mathrm{AD}\&comma;\mathrm{Typesetting}:-\mathrm{EV}\&comma;\mathrm{Typesetting}:-\mathrm{mi}\&comma;\mathrm{Typesetting}:-\mathrm{mio}\&comma;\mathrm{Typesetting}:-\mathrm{mn}\&comma;\mathrm{Typesetting}:-\mathrm{mo}\&comma;\mathrm{Typesetting}:-\mathrm{ms}\&comma;\mathrm{Typesetting}:-\mathrm{mtd}\&comma;\mathrm{Typesetting}:-\mathrm{mtr}\&comma;\mathrm{Typesetting}:-\mathrm{maction}\&comma;\mathrm{Typesetting}:-\mathrm{maddr}\&comma;\mathrm{Typesetting}:-\mathrm{matomic}\&comma;\mathrm{Typesetting}:-\mathrm{mempty}\&comma;\mathrm{Typesetting}:-\mathrm{merror}\&comma;\mathrm{Typesetting}:-\mathrm{mfail}\&comma;\mathrm{Typesetting}:-\mathrm{mfenced}\&comma;\mathrm{Typesetting}:-\mathrm{mfrac}\&comma;\mathrm{Typesetting}:-\mathrm{mglyph}\&comma;\mathrm{Typesetting}:-\mathrm{mover}\&comma;\mathrm{Typesetting}:-\mathrm{mpadded}\&comma;\mathrm{Typesetting}:-\mathrm{mparsed}\&comma;\mathrm{Typesetting}:-\mathrm{mprimed}\&comma;\mathrm{Typesetting}:-\mathrm{mroot}\&comma;\mathrm{Typesetting}:-\mathrm{mrow}\&comma;\mathrm{Typesetting}:-\mathrm{mspace}\&comma;\mathrm{Typesetting}:-\mathrm{msqrt}\&comma;\mathrm{Typesetting}:-\mathrm{mstyle}\&comma;\mathrm{Typesetting}:-\mathrm{msub}\&comma;\mathrm{Typesetting}:-\mathrm{msubsup}\&comma;\mathrm{Typesetting}:-\mathrm{msup}\&comma;\mathrm{Typesetting}:-\mathrm{mtable}\&comma;\mathrm{Typesetting}:-\mathrm{mtext}\&comma;\mathrm{Typesetting}:-\mathrm{munder}\&comma;\mathrm{Typesetting}:-\mathrm{none}\&comma;\mathrm{Typesetting}:-\mathrm{maligngroup}\&comma;\mathrm{Typesetting}:-\mathrm{malignmark}\&comma;\mathrm{Typesetting}:-\mathrm{mambiguous}\&comma;\mathrm{Typesetting}:-\mathrm{menclose}\&comma;\mathrm{Typesetting}:-\mathrm{mlabeledtr}\&comma;\mathrm{Typesetting}:-\mathrm{mpattern}\&comma;\mathrm{Typesetting}:-\mathrm{mphantom}\&comma;\mathrm{Typesetting}:-\mathrm{mprescripts}\&comma;\mathrm{Typesetting}:-\mathrm{mprintslash}\&comma;\mathrm{Typesetting}:-\mathrm{mscripts}\&comma;\mathrm{Typesetting}:-\mathrm{msemantics}\&comma;\mathrm{Typesetting}:-\mathrm{munderover}\&comma;\mathrm{Typesetting}:-\mathrm{mverbatim}\&comma;\mathrm{Typesetting}:-\mathrm{mmultiscripts}\right]\&comma;\mathrm{hypergeom}\right)$

Improved differentiation of special functions with respect to parameters:

Diff( MeijerG([[], [b+1, b+1]],[[b, b, f(a)], []],z), a );

$\frac{\&DifferentialD;}{\&DifferentialD;a}\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[b+1\&comma;b+1\right]\right]\&comma;\left[\left[f\left(a\right)\&comma;b\&comma;b\right]\&comma;\left[\right]\right]\&comma;z\right)$

value((60));

$\left(\ln \left(z\right)\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[b+1\&comma;b+1\right]\right]\&comma;\left[\left[f\left(a\right)\&comma;b\&comma;b\right]\&comma;\left[\right]\right]\&comma;z\right)+2\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[b+1\&comma;b+1\&comma;b+1\right]\right]\&comma;\left[\left[f\left(a\right)\&comma;b\&comma;b\&comma;b\right]\&comma;\left[\right]\right]\&comma;z\right)\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;a}f\left(a\right)\right)$

Diff( MeijerG([[], []],[[f(a)], []],g(z)), a );

$\frac{\&DifferentialD;}{\&DifferentialD;a}\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[\right]\right]\&comma;\left[\left[f\left(a\right)\right]\&comma;\left[\right]\right]\&comma;g\left(z\right)\right)$

value((62));

$\ln \left(g\left(z\right)\right)\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[\right]\right]\&comma;\left[\left[f\left(a\right)\right]\&comma;\left[\right]\right]\&comma;g\left(z\right)\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;a}f\left(a\right)\right)$

There are new differentiation rules for the argument, Re, Im, and conjugate functions. Although these functions do not strictly satisfy the Cauchy-Riemann conditions, in the same way that Maple implements a differentiation rule for abs and signum, Maple now also implements the differentiation rule for these functions in terms of the 'derivatives' of abs and signum. This allows further computation when these functions are present and differentiation is used. Maple correctly manipulates the results in terms of abs' and signum', which are represented by abs(1,z) and signum(1,z).

z = abs(z)*exp(I*argument(z));  # definition of a complex variable z

$z=\left|z\right|{\&ExponentialE;}^{\mathrm{I}\arg \left(z\right)}$

diff( (64), z);    # a 'representation' for d/dz argument(z) using abs(1,z)

$1=\mathrm{abs}\left(1\&comma;z\right){\&ExponentialE;}^{\mathrm{I}\arg \left(z\right)}+\mathrm{I}\left|z\right|\left(-\frac{\mathrm{I}}{z}+\frac{\mathrm{I}\mathrm{abs}\left(1\&comma;z\right)}{\left|z\right|}\right){\&ExponentialE;}^{\mathrm{I}\arg \left(z\right)}$

expand( (65) );    # in this example, abs(1,z) cancels out

$1=\frac{\left|z\right|{\&ExponentialE;}^{\mathrm{I}\arg \left(z\right)}}{z}$

simplify( (66) );  # further correct manipulations are then feasible

$1=1$

Combinations in Sums now take into account that the summation variable can have only integer values in the summation range.

Examples

ee := k*(2*ln(k+1)-ln(k)-ln(k+2));

$\mathrm{ee}≔k\left(2\ln \left(k+1\right)-\ln \left(k\right)-\ln \left(k+2\right)\right)$

combine(ee, ln);   # <- k is an arbitrary complex variable, nothing to do

$k\left(2\ln \left(k+1\right)-\ln \left(k\right)-\ln \left(k+2\right)\right)$

Sum( ee, k=1..infinity );   # k is now an integer from 1 to infinity

$\sum _{k=1}^{\mathrm{\infty }}k\left(2\ln \left(k+1\right)-\ln \left(k\right)-\ln \left(k+2\right)\right)$

combine( (70), ln );   # a combination is therefore possible

$\sum _{k=1}^{\mathrm{\infty }}k\ln \left(\frac{{\left(k+1\right)}^2}{k\left(k+2\right)}\right)$

The combination of logarithms is now more general.

-ln(5) + ln(abs(x)) - ln(x);

$-\ln \left(5\right)+\ln \left(\left|x\right|\right)-\ln \left(x\right)$

combine((72));

$-\ln \left(\frac{5x}{\left|x\right|}\right)$

ln(x)-ln(y);

$\ln \left(x\right)-\ln \left(y\right)$

combine((74)) assuming x > 0;

$-\ln \left(\frac{y}{x}\right)$

ln(a-1)+I*Pi-ln(1-a);

$\ln \left(a-1\right)+\mathrm{I}\mathrm{\pi }-\ln \left(1-a\right)$

combine((76)) assuming a > 1;

$0$

The series command can compute series expansions for more functions.

Examples

EllipticModulus(q);

$\mathrm{EllipticModulus}\left(q\right)$

series((78), q);

$4\sqrt{q}-16q^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+56q^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-160q^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+404q^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-944q^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\mathrm{O}\left(q^{\raisebox{1ex}{$13$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$

As a result of the change in the form of the differentiation rule for GAMMA(a,z) and Ei(a,z), it is now possible to also compute series expansions with respect to the first parameter $a$.

GAMMA(a,z);

$\mathrm{\Gamma }\left(a\&comma;z\right)$

series((80), a, 2);

${\mathrm{Ei}}_1\left(z\right)+\left({\mathrm{Ei}}_1\left(z\right)\ln \left(z\right)+\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[1\&comma;1\right]\right]\&comma;\left[\left[0\&comma;0\&comma;0\right]\&comma;\left[\right]\right]\&comma;z\right)\right)a+O\left(a^2\right)$

Ei(a,z);

${\mathrm{Ei}}_a\left(z\right)$

series((82), a, 2);

$\frac{{\&ExponentialE;}^{-z}}{z}-\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[0\right]\right]\&comma;\left[\left[\mathrm{-1}\&comma;\mathrm{-1}\right]\&comma;\left[\right]\right]\&comma;z\right)a+O\left(a^2\right)$

A new option in plots[plotcompare], shift_range = r + s I, is available to shift the range of the plot, for example, from $\left(a+bI\right)\mathrm{..}\left(c+dI\right)$ to $\left(a+r+\left(b+s\right)I\right)\mathrm{..}\left(c+r+\left(d+s\right)I\right)$. This is useful when the difference between the expressions plotted is visible only after shifting the plotting range, for example, when comparing a function with its series representation.

Example

A comparison of $\sin \left(x\right)$ with its cubic Taylor polynomial.

plots[plotcompare]( sin(x), x-x^3/6, x=-3..3 );

If you shift the range, for example, to the right by 10, the function and its cubic approximation no longer look similar.

plots[plotcompare]( sin(x), x-x^3/6, x=-3..3, shift_range=10 );

The new FunctionDecomposition command in the DEtools package performs rational function decomposition on overdetermined systems with parameters. This type of function decomposition can be used to solve symbolic problems in various areas. The command has natural application beyond differential equations.

The singular command now accepts a numeric range as an extra argument, which causes only the singularities within this range to be returned.

Example

singular( tan(x) );         # works as in previous releases

$\left\{x=\frac{1}{2}\mathrm{\pi }+\mathrm{\_Z1~}\mathrm{\pi }\right\}$

singular( tan(x), 1..10 );  # new

$\left\{x=\frac{\mathrm{\pi }}{2}\right\},\left\{x=\frac{3\mathrm{\pi }}{2}\right\},\left\{x=\frac{5\mathrm{\pi }}{2}\right\}$