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-{"documenter":{"julia_version":"1.11.1","generation_timestamp":"2024-11-04T13:30:26","documenter_version":"1.7.0"}}
+{"documenter":{"julia_version":"1.11.1","generation_timestamp":"2024-11-05T01:36:41","documenter_version":"1.7.0"}}
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Taylor Series · Symbolics.jl</title><meta name="title" content="Taylor Series · Symbolics.jl"/><meta property="og:title" content="Taylor Series · Symbolics.jl"/><meta property="twitter:title" content="Taylor Series · Symbolics.jl"/><meta name="description" content="Documentation for Symbolics.jl."/><meta property="og:description" content="Documentation for Symbolics.jl."/><meta property="twitter:description" content="Documentation for Symbolics.jl."/><meta property="og:url" content="https://docs.sciml.ai/Symbolics/stable/manual/taylor/"/><meta property="twitter:url" content="https://docs.sciml.ai/Symbolics/stable/manual/taylor/"/><link rel="canonical" href="https://docs.sciml.ai/Symbolics/stable/manual/taylor/"/><script data-outdated-warner src="../../assets/warner.js"></script><link 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src="../../assets/themeswap.js"></script><link href="../../assets/favicon.ico" rel="icon" type="image/x-icon"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img class="docs-light-only" src="../../assets/logo.svg" alt="Symbolics.jl logo"/><img class="docs-dark-only" src="../../assets/logo-dark.svg" alt="Symbolics.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../../">Symbolics.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><a class="tocitem" href="../../getting_started/">Getting Started with Symbolics.jl</a></li><li><span class="tocitem">Tutorials</span><ul><li><a class="tocitem" href="../../tutorials/auto_parallel/">Automated Sparse Parallelism of Julia Functions via Tracing</a></li><li><a class="tocitem" href="../../tutorials/converting_to_C/">Automatic Conversion of Julia Code to C Functions</a></li><li><a class="tocitem" href="../../tutorials/perturbation/">Mixed Symbolic-Numeric Perturbation Theory</a></li></ul></li><li><span class="tocitem">Manual</span><ul><li><a class="tocitem" href="../variables/">Variable and Equation Types</a></li><li><a class="tocitem" href="../expression_manipulation/">Expression Manipulation</a></li><li><a class="tocitem" href="../derivatives/">Derivatives and Differentials</a></li><li class="is-active"><a class="tocitem" href>Taylor Series</a></li><li><a class="tocitem" href="../groebner/">Groebner bases</a></li><li><a class="tocitem" href="../solver/">Solver</a></li><li><a class="tocitem" href="../arrays/">Symbolic Arrays</a></li><li><a class="tocitem" href="../build_function/">Function Building and Compilation (build_function)</a></li><li><a class="tocitem" href="../functions/">Function Registration and Tracing</a></li><li><a class="tocitem" href="../parsing/">Parsing Julia Expressions to Symbolic Expressions</a></li><li><a class="tocitem" href="../io/">I/O, Saving, and Latex</a></li><li><a class="tocitem" href="../sparsity_detection/">Structure and Sparsity Detection</a></li><li><a class="tocitem" href="../types/">Supported types and dispatch in Symbolics</a></li><li><a class="tocitem" href="../faq/">Frequently Asked Questions</a></li><li><a class="tocitem" href="../limits/">Symbolic Limits</a></li></ul></li><li><a class="tocitem" href="../../comparison/">Comparison Against SymPy</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Taylor Series</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Taylor Series</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/JuliaSymbolics/Symbolics.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/JuliaSymbolics/Symbolics.jl/blob/master/docs/src/manual/taylor.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Taylor-Series"><a class="docs-heading-anchor" href="#Taylor-Series">Taylor Series</a><a id="Taylor-Series-1"></a><a class="docs-heading-anchor-permalink" href="#Taylor-Series" title="Permalink"></a></h1><p>For a real example of how to use the Taylor series functionality, see <a href="../../tutorials/perturbation/#perturb_alg">this tutorial</a>.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Symbolics.series" href="#Symbolics.series"><code>Symbolics.series</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">series(cs, x, [x0=0,], ns=0:length(cs)-1)</code></pre><p>Return the power series in <code>x</code> around <code>x0</code> to the powers <code>ns</code> with coefficients <code>cs</code>.</p><pre><code class="nohighlight hljs">series(y, x, [x0=0,] ns)</code></pre><p>Return the power series in <code>x</code> around <code>x0</code> to the powers <code>ns</code> with coefficients automatically created from the variable <code>y</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">julia&gt; @variables x y[0:3] z
+3-element Vector{Any}:
+ x
+  y[0:3]
+ z
+
+julia&gt; series(y, x, 2)
+y[0] + (-2 + x)*y[1] + ((-2 + x)^2)*y[2] + ((-2 + x)^3)*y[3]
+
+julia&gt; series(z, x, 2, 0:3)
+z[0] + (-2 + x)*z[1] + ((-2 + x)^2)*z[2] + ((-2 + x)^3)*z[3]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaSymbolics/Symbolics.jl/blob/1305b7ec94d5d81f73b4162737c8fc84f1ae21e9/src/taylor.jl#L1-L26">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Symbolics.taylor" href="#Symbolics.taylor"><code>Symbolics.taylor</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">taylor(f, x, [x0=0,] n; rationalize=true)</code></pre><p>Calculate the <code>n</code>-th order term(s) in the Taylor series of <code>f</code> around <code>x = x0</code>. If <code>rationalize</code>, float coefficients are approximated as rational numbers (this can produce unexpected results for irrational numbers, for example).</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">julia&gt; @variables x
+1-element Vector{Num}:
+ x
+
+julia&gt; taylor(exp(x), x, 0:3)
+1 + x + (1//2)*(x^2) + (1//6)*(x^3)
+
+julia&gt; taylor(exp(x), x, 0:3; rationalize=false)
+1.0 + x + 0.5(x^2) + 0.16666666666666666(x^3)
+
+julia&gt; taylor(√(x), x, 1, 0:3)
+1 + (1//2)*(-1 + x) - (1//8)*((-1 + x)^2) + (1//16)*((-1 + x)^3)
+
+julia&gt; isequal(taylor(exp(im*x), x, 0:5), taylor(exp(im*x), x, 0:5))
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaSymbolics/Symbolics.jl/blob/1305b7ec94d5d81f73b4162737c8fc84f1ae21e9/src/taylor.jl#L96-L121">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Symbolics.taylor_coeff" href="#Symbolics.taylor_coeff"><code>Symbolics.taylor_coeff</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">taylor_coeff(f, x[, n]; rationalize=true)</code></pre><p>Calculate the <code>n</code>-th order coefficient(s) in the Taylor series of <code>f</code> around <code>x = 0</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">julia&gt; @variables x y
+2-element Vector{Num}:
+ x
+ y
+
+julia&gt; taylor_coeff(series(y, x, 0:5), x, 0:2:4)
+3-element Vector{Num}:
+ y[0]
+ y[2]
+ y[4]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaSymbolics/Symbolics.jl/blob/1305b7ec94d5d81f73b4162737c8fc84f1ae21e9/src/taylor.jl#L43-L62">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../derivatives/">« Derivatives and Differentials</a><a class="docs-footer-nextpage" href="../groebner/">Groebner bases »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Tuesday 5 November 2024 01:36">Tuesday 5 November 2024</span>. Using Julia version 1.11.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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		{"documenter":{"julia_version":"1.11.1","generation_timestamp":"2024-11-05T01:36:41","documenter_version":"1.7.0"}}