Zoom - Maple Help

MathematicalFunctions[Evalf]

 Zoom
 zoom within the last plot computed using 'Evalf' related to concatenated Taylor expansions used when performing numerical computations of Heun or Appell functions

 Calling Sequence Zoom() Zoom([n, m])

Parameters

 [n, m] - (optional) list with two numbers, where $n$ represents the origin of the returned plot and $m$ indicates the width to the left and to the right of $n$ where the plot is being zoomed. The default value is [1, 1].

Description

 • The Zoom command is used to zoom within the last plot computed using Evalf using a concatenated Taylor expansions approach when performing numerical evaluations of Heun or Appell functions. This command is useful to understand how the method of a sequence of concatenated Taylor expansions work.
 • When called with the extra argument [n, m], $n$ represents the origin of the returned plot and $m$ indicates the width to the left and to the right of $n$ where the plot is being zoomed. Calling Zoom with no arguments is the same as calling it with the optional argument [1, 1].

Examples

 Initialization: Load the command and package and set the display of special functions in output to typeset mathematical notation (textbook notation):
 > $\mathrm{with}\left(\mathrm{MathematicalFunctions},\mathrm{Evalf}\right)$
 $\left[{\mathrm{Evalf}}\right]$ (1)
 > $\mathrm{with}\left(\mathrm{MathematicalFunctions}:-\mathrm{Evalf}\right);\mathrm{Typesetting}:-\mathrm{EnableTypesetRule}\left(\mathrm{Typesetting}:-\mathrm{SpecialFunctionRules}\right):$
 $\left\{{\mathrm{Add}}{,}{\mathrm{Evalb}}{,}{\mathrm{Zoom}}{,}{\mathrm{QuadrantNumbers}}{,}{\mathrm{Singularities}}{,}{\mathrm{GenerateRecurrence}}{,}{\mathrm{PairwiseSummation}}\right\}$ (2)

Consider one of the special values of AppellF1, a case where the function can be represented by a 2F1 hypergeometric function

 >
 ${\mathrm{F1}}{≔}{\mathrm{%AppellF1}}{}\left({4.0}{,}{2.0}{,}{0.3}{,}{2.3}{,}{1.12}{,}{1.1}\right)$ (3)
 > $\mathrm{F1}=\mathrm{value}\left(\mathrm{convert}\left(\mathrm{F1},\mathrm{rational}\right)\right)$
 ${\mathrm{%AppellF1}}{}\left({4.0}{,}{2.0}{,}{0.3}{,}{2.3}{,}{1.12}{,}{1.1}\right){=}{10000}{}{}_{{2}}{F}_{{1}}{}\left({2}{,}{4}{;}\frac{{23}}{{10}}{;}{-}\frac{{1}}{{5}}\right)$ (4)

The left-hand side is AppellF1 in inert form, to avoid the automatic representation in terms of 2F1 functions, while the right-hand side involves only a hypergeometric 2F1 function. Evaluate this expression numerically

 > $\mathrm{evalf}\left(\right)$
 ${5325.710910}{=}{5325.710910}$ (5)

Compute the same but now using a concatenated Taylor series expansion, and displaying a plot showing the centers and path of the Taylor expansions used and no other information

 > $\mathrm{Evalf}\left(,\mathrm{usetaylor},\mathrm{time},\mathrm{plot},\mathrm{quiet}\right)$
 ${\mathrm{CPU time elapsed during evaluation: .78e-1 seconds}}$
 ${5325.710910}{+}{1.}{}{{10}}^{{-13}}{}{I}{=}{5325.710910}$ (6)

Zoom closer to the evaluation point $1.1$, extending 1/50 to the left and right of $1.1$

 > $\mathrm{Zoom}\left(\left[1.1,\frac{1}{50}\right]\right)$

In this Taylor approach, each expansion around a point is used to reach up to 95/100 of the radius of convergence before starting another expansion. Reduce that to 1/2, compute internally at Digits = 50 (but return as if computing with Digits = 10)

 > $\mathrm{Evalf}\left(,50,\mathrm{usetaylor},R=\frac{1}{2},\mathrm{time},\mathrm{plot},\mathrm{zoom}=\left[1,\frac{1}{2}\right],\mathrm{quiet}\right)$
 ${\mathrm{CPU time elapsed during evaluation: .281 seconds}}$
 ${5325.710910}{+}{1.410240821}{}{{10}}^{{-52}}{}{I}{=}{5325.710910}$ (7)

Use Zoom to zoom closer to the point $z=1.1$

 > $\mathrm{Zoom}\left(\left[1.1,\frac{1}{15}\right]\right)$

Compatibility

 • The MathematicalFunctions[Evalf][Zoom] command was introduced in Maple 2017.