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4<html>
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6    <meta charset="utf-8" />
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8<title>Sage Reference &#8212; Sage Reference v9.2.beta9</title>
9
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19    <link rel="index" title="Index" href="genindex.html" />
20    <link rel="search" title="Search" href="search.html" />
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30  </head><body>
31    <div class="related" role="navigation" aria-label="related navigation">
32      <h3>Navigation</h3>
33      <ul>
34        <li class="right" style="margin-right: 10px">
35          <a href="genindex.html" title="General Index"
36             accesskey="I">index</a></li>
37 
38    <a href="/Users/kwankyu/GitHub/sage-dev/local/share/doc/sage/html/en/index.html"><img src="_static/logo_sagemath_black.svg" height="28" style="vertical-align: middle" title="Sage Logo"></a>
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40 
41        <li class="nav-item nav-item-0"><a href="#">Sage Reference v9.2.beta9</a> &#187;</li>
42
43        <li class="nav-item nav-item-this"><a href=""></a></li> 
44      </ul>
45    </div> 
46
47    <div class="document">
48      <div class="documentwrapper">
49        <div class="bodywrapper">
50          <div class="body" role="main">
51           
52<h1> Sage Reference Manual (PDF version)</h1></p>
53<p>Welcome to the Sage reference manual. Here you find documentation for all of
54<a href="http://www.sagemath.org/">Sage</a>'s features, illustrated with lots of examples.
55A thematic index follows.</p>
56<p>This documentation is licensed under the <a href="http://creativecommons.org/licenses/by-sa/3.0/">Creative Commons Attribution-Share Alike 3.0 License</a>.
57<ul></ul>
58
59
60<h2>User Interfaces</h2>
61
62<ul>
63
64<li> <a href="repl/repl.pdf">Command Line Interface <img src="_static/pdf.png"/></a></li>
65<li> <a href="https://jupyter-notebook.readthedocs.io/en/latest/notebook.html">Jupyter Notebook Interface</a></li>
66</ul>
67
68
69<h2>Graphics</h2>
70
71<ul>
72
73<li> <a href="plotting/plotting.pdf">2D Graphics <img src="_static/pdf.png"/></a></li>
74<li> <a href="plot3d/plot3d.pdf">3D Graphics <img src="_static/pdf.png"/></a></li>
75</ul>
76
77
78<h2>Mathematics</h2>
79
80<ul>
81</ul>
82
83
84<h3>Parents and Categories</h3>
85
86<ul>
87
88<li> <a href="structure/structure.pdf">Parents and Elements <img src="_static/pdf.png"/></a></li>
89<li> <a href="coercion/coercion.pdf">Coercion <img src="_static/pdf.png"/></a></li>
90<li> <a href="categories/categories.pdf">Categories <img src="_static/pdf.png"/></a></li>
91</ul>
92
93
94<h3>Basic Rings and Fields</h3>
95
96<ul>
97
98<li> <a href="rings_standard/rings_standard.pdf">Integers and Rational Numbers <img src="_static/pdf.png"/></a></li>
99<li> <a href="rings_numerical/rings_numerical.pdf">Real and Complex Numbers <img src="_static/pdf.png"/></a></li>
100<li> <a href="polynomial_rings/polynomial_rings.pdf">Commutative Polynomials <img src="_static/pdf.png"/></a></li>
101<li> <a href="power_series/power_series.pdf">Power Series and Laurent Series <img src="_static/pdf.png"/></a></li>
102<li> <a href="finite_rings/finite_rings.pdf">Finite Rings and Fields <img src="_static/pdf.png"/></a></li>
103<li> <a href="padics/padics.pdf">\(p\)-adic Numbers <img src="_static/pdf.png"/></a></li>
104<li> <a href="noncommutative_polynomial_rings/noncommutative_polynomial_rings.pdf">Noncommutative Polynomials <img src="_static/pdf.png"/></a></li>
105<li> <a href="quat_algebras/quat_algebras.pdf">Quaternion Algebras <img src="_static/pdf.png"/></a></li>
106</ul>
107
108
109<h3>Linear Algebra</h3>
110
111<ul>
112
113<li> <a href="matrices/matrices.pdf">Matrices and Spaces of Matrices <img src="_static/pdf.png"/></a></li>
114<li> <a href="modules/modules.pdf">Vectors and Modules <img src="_static/pdf.png"/></a></li>
115<li> <a href="tensor_free_modules/tensor_free_modules.pdf">Tensors on Free Modules of Finite Rank <img src="_static/pdf.png"/></a></li>
116</ul>
117
118
119<h3>Calculus and Analysis</h3>
120
121<ul>
122
123<li> <a href="calculus/calculus.pdf">Symbolic Calculus <img src="_static/pdf.png"/></a></li>
124<li> <a href="constants/constants.pdf">Mathematical Constants <img src="_static/pdf.png"/></a></li>
125<li> <a href="functions/functions.pdf">Elementary and Special Functions <img src="_static/pdf.png"/></a></li>
126<li> <a href="asymptotic/asymptotic.pdf">Asymptotic Expansions <img src="_static/pdf.png"/></a></li>
127<li> <a href="numerical/numerical.pdf">Numerical Optimization <img src="_static/pdf.png"/></a></li>
128</ul>
129
130
131<h3>Probability and Statistics</h3>
132
133<ul>
134
135<li> <a href="probability/probability.pdf">Probability <img src="_static/pdf.png"/></a></li>
136<li> <a href="stats/stats.pdf">Statistics <img src="_static/pdf.png"/></a></li>
137<li> <a href="finance/finance.pdf">Quantitative Finance <img src="_static/pdf.png"/></a></li>
138</ul>
139
140
141<h3>Mathematical Structures</h3>
142
143<ul>
144
145<li> <a href="sets/sets.pdf">Sets <img src="_static/pdf.png"/></a></li>
146<li> <a href="monoids/monoids.pdf">Monoids <img src="_static/pdf.png"/></a></li>
147<li> <a href="groups/groups.pdf">Groups <img src="_static/pdf.png"/></a></li>
148<li> <a href="semirings/semirings.pdf">Semirings <img src="_static/pdf.png"/></a></li>
149<li> <a href="rings/rings.pdf">Rings <img src="_static/pdf.png"/></a></li>
150<li> <a href="algebras/algebras.pdf">Algebras <img src="_static/pdf.png"/></a></li>
151</ul>
152
153
154<h3>Discrete Mathematics</h3>
155
156<ul>
157
158<li> <a href="combinat/combinat.pdf">Combinatorics <img src="_static/pdf.png"/></a></li>
159<li> <a href="graphs/graphs.pdf">Graph Theory <img src="_static/pdf.png"/></a></li>
160<li> <a href="quivers/quivers.pdf">Quivers <img src="_static/pdf.png"/></a></li>
161<li> <a href="matroids/matroids.pdf">Matroid Theory <img src="_static/pdf.png"/></a></li>
162<li> <a href="dynamics/dynamics.pdf">Discrete Dynamics <img src="_static/pdf.png"/></a></li>
163<li> <a href="coding/coding.pdf">Coding Theory <img src="_static/pdf.png"/></a></li>
164<li> <a href="cryptography/cryptography.pdf">Cryptography <img src="_static/pdf.png"/></a></li>
165<li> <a href="game_theory/game_theory.pdf">Game Theory <img src="_static/pdf.png"/></a></li>
166<li> <a href="logic/logic.pdf">Symbolic Logic <img src="_static/pdf.png"/></a></li>
167<li> <a href="sat/sat.pdf">SAT solvers <img src="_static/pdf.png"/></a></li>
168</ul>
169
170
171<h3>Geometry and Topology</h3>
172
173<ul>
174
175<li> <a href="euclidean_spaces/euclidean_spaces.pdf">Euclidean Spaces and Vector Calculus <img src="_static/pdf.png"/></a></li>
176<li> <a href="discrete_geometry/discrete_geometry.pdf">Combinatorial and Discrete Geometry <img src="_static/pdf.png"/></a></li>
177<li> <a href="homology/homology.pdf">Cell Complexes and their Homology <img src="_static/pdf.png"/></a></li>
178<li> <a href="manifolds/manifolds.pdf">Manifolds and Differential Geometry <img src="_static/pdf.png"/></a></li>
179<li> <a href="hyperbolic_geometry/hyperbolic_geometry.pdf">Hyperbolic Geometry <img src="_static/pdf.png"/></a></li>
180<li> <a href="riemannian_geometry/riemannian_geometry.pdf">Parametrized Surfaces <img src="_static/pdf.png"/></a></li>
181<li> <a href="knots/knots.pdf">Knot Theory <img src="_static/pdf.png"/></a></li>
182</ul>
183
184
185<h3>Number Fields, Function Fields, and Valuations</h3>
186
187<ul>
188
189<li> <a href="number_fields/number_fields.pdf">Number Fields <img src="_static/pdf.png"/></a></li>
190<li> <a href="function_fields/function_fields.pdf">Function Fields <img src="_static/pdf.png"/></a></li>
191<li> <a href="valuations/valuations.pdf">Discrete Valuations <img src="_static/pdf.png"/></a></li>
192</ul>
193
194
195<h3>Number Theory</h3>
196
197<ul>
198
199<li> <a href="diophantine_approximation/diophantine_approximation.pdf">Diophantine approximation <img src="_static/pdf.png"/></a></li>
200<li> <a href="quadratic_forms/quadratic_forms.pdf">Quadratic Forms <img src="_static/pdf.png"/></a></li>
201<li> <a href="lfunctions/lfunctions.pdf">\(L\)-Functions <img src="_static/pdf.png"/></a></li>
202<li> <a href="arithgroup/arithgroup.pdf">Arithmetic Subgroups of \({\rm SL}_2(\ZZ)\) <img src="_static/pdf.png"/></a></li>
203<li> <a href="hecke/hecke.pdf">General Hecke Algebras and Hecke Modules <img src="_static/pdf.png"/></a></li>
204<li> <a href="modsym/modsym.pdf">Modular Symbols <img src="_static/pdf.png"/></a></li>
205<li> <a href="modfrm/modfrm.pdf">Modular Forms <img src="_static/pdf.png"/></a></li>
206<li> <a href="modfrm_hecketriangle/modfrm_hecketriangle.pdf">Modular Forms for Hecke Triangle Groups <img src="_static/pdf.png"/></a></li>
207<li> <a href="modabvar/modabvar.pdf">Modular Abelian Varieties <img src="_static/pdf.png"/></a></li>
208<li> <a href="modmisc/modmisc.pdf">Miscellaneous Modular-Form-Related Modules <img src="_static/pdf.png"/></a></li>
209</ul>
210
211
212<h3>Algebraic and Arithmetic Geometry</h3>
213
214<ul>
215<li> <a href="schemes/schemes.pdf">Schemes <img src="_static/pdf.png"/></a></li>
216<li> <a href="curves/curves.pdf">Plane and Space Curves <img src="_static/pdf.png"/></a></li>
217<li> <a href="arithmetic_curves/arithmetic_curves.pdf">Elliptic and Hyperelliptic Curves <img src="_static/pdf.png"/></a></li>
218</ul>
219
220
221<h3>Miscellaneous</h3>
222
223<ul>
224
225<li> <a href="databases/databases.pdf">Databases <img src="_static/pdf.png"/></a></li>
226<li> <a href="games/games.pdf">Games <img src="_static/pdf.png"/></a></li>
227</ul>
228
229
230<h2>Programming</h2>
231
232<ul>
233</ul>
234
235
236<h3>Facilities</h3>
237
238<ul>
239
240<li> <a href="data_structures/data_structures.pdf">Data Structures <img src="_static/pdf.png"/></a></li>
241<li> <a href="misc/misc.pdf">Utilities <img src="_static/pdf.png"/></a></li>
242<li> <a href="doctest/doctest.pdf">Test Framework <img src="_static/pdf.png"/></a></li>
243<li> <a href="parallel/parallel.pdf">Parallel Computing <img src="_static/pdf.png"/></a></li>
244</ul>
245
246
247<h3>Interfaces</h3>
248
249<ul>
250
251<li> <a href="interfaces/interfaces.pdf">Interpreter Interfaces <img src="_static/pdf.png"/></a></li>
252<li> <a href="libs/libs.pdf">C/C++ Library Interfaces <img src="_static/pdf.png"/></a></li>
253<li> <a href="cpython/cpython.pdf">Python Technicalities <img src="_static/pdf.png"/></a></li>
254</ul>
255
256
257<h2>General Information</h2>
258
259<ul>
260
261<li> <a href="spkg/spkg.pdf">External Packages <img src="_static/pdf.png"/></a></li>
262<li> <a href="references/references.pdf">Bibliographic References <img src="_static/pdf.png"/></a></li>
263<li> <a href="history_and_license/history_and_license.pdf">History and License <img src="_static/pdf.png"/></a></li>
264
265</ul>
266
267
268
269
270
271
272            <div class="clearer"></div>
273          </div>
274        </div>
275      </div>
276
277      <div class="clearer"></div>
278    </div>
279    <div class="related" role="navigation" aria-label="related navigation">
280      <h3>Navigation</h3>
281      <ul>
282        <li class="right" style="margin-right: 10px">
283          <a href="genindex.html" title="General Index"
284             >index</a></li>
285 
286    <a href="/Users/kwankyu/GitHub/sage-dev/local/share/doc/sage/html/en/index.html"><img src="_static/logo_sagemath_black.svg" height="28" style="vertical-align: middle" title="Sage Logo"></a>
287 
288 
289        <li class="nav-item nav-item-0"><a href="#">Sage Reference v9.2.beta9</a> &#187;</li>
290
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292      </ul>
293    </div>
294   
295    <div class="footer" role="contentinfo">
296        &#169; Copyright 2005--2020, The Sage Development Team.
297      Created using <a href="https://www.sphinx-doc.org/">Sphinx</a> 3.1.2.
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