[{"id":"15b0b94c-34cc-44d9-bcfc-0fa59ffa054b","thread_id":"cff6883c-9c74-44cf-959a-57c7a7d83d21","created_at":"2025-06-13T18:50:16Z","branch_list":["00000000-0000-4000-0000-000000000000","f9ecef06-3738-4126-8f32-2cf5add029a0"],"state":"done","prompt":"I am reading that if we demand that a function f(x) is extendable to a function f(z) of the complex variable z such that f(z) is smooth in the sense that it is merely required to be once differentiable, then what does the word \"extendable\" mean in this context?","reply":"<p>In the context of extending a real-valued function <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> to a complex-valued function <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span> that is once differentiable with respect to the complex variable <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi></mrow></math></span>, \"extendable\" means that there exists a function <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span> defined on an open set <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>Ω</mi></mrow></math></span> in the complex plane such that:</p>\n<ol>\n<li>The set <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>Ω</mi></mrow></math></span> contains the domain (or a portion of it) of the original real function <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> as a subset of the real axis.</li>\n<li>For all real numbers <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi></mrow></math></span> in the domain of <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi></mrow></math></span> that are also in <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>Ω</mi></mrow></math></span>, the complex function <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span> evaluated at <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi><mo>=</mo><mi>x</mi></mrow></math></span> (i.e., <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mn>0</mn><mi>i</mi></mrow></math></span>) is equal to <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span>. So, <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span>.</li>\n<li>The function <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span> is complex differentiable (holomorphic) at every point in the open set <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>Ω</mi></mrow></math></span>. </li>\n</ol>\n<p>A key point in complex analysis is that if a function is once complex differentiable in an open set, it is automatically infinitely differentiable (analytic) in that set. This is a much stronger condition than for real functions, where a function can be once differentiable without being twice differentiable.</p>\n<p>Therefore, for <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> to be \"extendable\" in this manner, <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> must be a real analytic function on its domain. This means that <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> can be represented by a convergent power series (Taylor series) in a neighborhood of each point in its domain. </p>\n<p>For example:</p>\n<ul>\n<li>The real function <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> can be extended to the complex function <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msup><mi>z</mi><mn>2</mn></msup></mrow></math></span>, which is differentiable everywhere in the complex plane.</li>\n<li>The real function <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup></mrow></math></span> can be extended to <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msup><mi>e</mi><mi>z</mi></msup></mrow></math></span>, which is differentiable everywhere.</li>\n<li>However, a function like <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo stretchy=\"false\">|</mo><mi>x</mi><mo stretchy=\"false\">|</mo></mrow></math></span> (absolute value) is not differentiable at <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span> and cannot be extended to a holomorphic function in any neighborhood of <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mn>0</mn></mrow></math></span>. Even a smooth real function like <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></mrow></math></span> for <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow></math></span> and <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mn>0</mn></mrow></math></span>, which is infinitely differentiable on the real line, is not real analytic at <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span> and thus cannot be extended to a function holomorphic in a neighborhood of <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mn>0</mn></mrow></math></span>. </li>\n</ul>","md":"In the context of extending a real-valued function $f(x)$ to a complex-valued function $f(z)$ that is once differentiable with respect to the complex variable $z$, \"extendable\" means that there exists a function $F(z)$ defined on an open set $\\Omega$ in the complex plane such that:\n1. The set $\\Omega$ contains the domain (or a portion of it) of the original real function $f(x)$ as a subset of the real axis.\n2. For all real numbers $x$ in the domain of $f$ that are also in $\\Omega$, the complex function $F(z)$ evaluated at $z=x$ (i.e., $z=x+0i$) is equal to $f(x)$. So, $F(x) = f(x)$.\n3. The function $F(z)$ is complex differentiable (holomorphic) at every point in the open set $\\Omega$. \n\nA key point in complex analysis is that if a function is once complex differentiable in an open set, it is automatically infinitely differentiable (analytic) in that set. This is a much stronger condition than for real functions, where a function can be once differentiable without being twice differentiable.\n\nTherefore, for $f(x)$ to be \"extendable\" in this manner, $f(x)$ must be a real analytic function on its domain. This means that $f(x)$ can be represented by a convergent power series (Taylor series) in a neighborhood of each point in its domain. \n\nFor example:\n- The real function $f(x) = x^2$ can be extended to the complex function $F(z) = z^2$, which is differentiable everywhere in the complex plane.\n- The real function $f(x) = e^x$ can be extended to $F(z) = e^z$, which is differentiable everywhere.\n- However, a function like $f(x) = |x|$ (absolute value) is not differentiable at $x=0$ and cannot be extended to a holomorphic function in any neighborhood of $0$. Even a smooth real function like $f(x) = e^{-1/x^2}$ for $x \\neq 0$ and $f(0)=0$, which is infinitely differentiable on the real line, is not real analytic at $x=0$ and thus cannot be extended to a function holomorphic in a neighborhood of $0$. ","profile":{"id":null,"name":null,"model":"gemini-2-5-pro","model_name":"Gemini 2.5 Pro","model_provider":"google","model_input_limit":1000000,"recommended":false,"model_info":"\n\n<div class=\"heading\">\n <div class=\"left\">\n \n \n \n <svg class=\"model-icon icon-xs\" data-model=\"gemini-2-5-pro\">\n <use href=\"#icon-google\"/>\n </svg>\n <h3>\n \n <span class=\"model-provider\">Google</span>\n \n <span class=\"model-name\">none</span>\n </h3>\n </div>\n\n \n <a href=\"/settings?p=custom_assistant&id=none\">\n <svg class=\"icon-sm edit-icon\" aria-hidden=\"true\">\n <use href=\"#icon-pencil\" />\n </svg>\n </a>\n \n</div>\n\n\n<hr />\n<ul>\n \n <li>\n <h4>\n Cost\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Relatively expensive\">\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --medium\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --medium\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --medium\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --medium\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-cost --medium\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Quality\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 4 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Speed\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 2 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Privacy\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 4 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li class=\"divider\" aria-hidden=\"true\"></li>\n <li>\n <h4>Last updated</h4>\n <span class=\"date\">\n \n \n 2025-06-17\n \n \n </span>\n </li>\n \n</ul>\n","model_provider_name":"Google","internet_access":true,"personalizations":true,"shortcut":null,"is_default_profile":false},"metadata":"<li>\n <span class=\"attribute\">Model</span>\n <span class=\"value\"><i class=\"icon-xs mr-2\"><svg><use href=\"#icon-google\"></use></svg></i>Gemini 2.5 Pro Preview</span>\n</li>\n<li>\n <span class=\"attribute\">Version</span>\n <span class=\"value\">models/gemini-2.5-pro-preview-05-06</span>\n</li>\n<li>\n <span class=\"attribute\">Speed (tok/s)</span>\n <span class=\"value\">30</span>\n</li>\n<li>\n <span class=\"attribute\">Tokens</span>\n <span class=\"value\">1,341</span>\n</li>\n<li>\n <span class=\"attribute\">Cost / Total ($)</span>\n <span class=\"value\">0.008 / 0.008</span>\n</li>\n<li>\n <span class=\"attribute\">End to end time (s)</span>\n <span class=\"value\">17.88</span>\n</li>\n\n","documents":[]},{"id":"bf4e00c1-5584-4e0c-97eb-82f9adfa9767","thread_id":"cff6883c-9c74-44cf-959a-57c7a7d83d21","created_at":"2025-06-13T19:05:16Z","state":"done","prompt":"So if we use a function like f(x) equals 1/x, and then extend it to f(z) equals 1/z, what does the function look like in terms of both its initial shape as well as its differentiability once?","reply":"<p>Okay, let's break down <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mi>x</mi></mrow></math></span> and its extension <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mi>z</mi></mrow></math></span>.</p>\n<p><strong>1. The Real Function: <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mi>x</mi></mrow></math></span></strong></p>\n<ul>\n<li>\n<p><strong>Initial Shape (Graph):</strong></p>\n<ul>\n<li>This function describes a <strong>hyperbola</strong>.</li>\n<li>It is defined for all real numbers <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi></mrow></math></span> except <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span>.</li>\n<li>For <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow></math></span>: As <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi></mrow></math></span> gets very small (approaches 0 from the positive side), <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> becomes very large and positive (approaches <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mo>+</mo><mo>∞</mo></mrow></math></span>). As <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi></mrow></math></span> gets very large, <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> gets very small and positive (approaches 0).</li>\n<li>For <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi><mo><</mo><mn>0</mn></mrow></math></span>: As <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi></mrow></math></span> gets very small in magnitude (approaches 0 from the negative side), <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> becomes very large and negative (approaches <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mo>−</mo><mo>∞</mo></mrow></math></span>). As <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi></mrow></math></span> gets very large in magnitude (but negative), <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> gets very small and negative (approaches 0).</li>\n<li>The graph has two separate branches, one in the first quadrant and one in the third quadrant. The x-axis and y-axis are asymptotes.</li>\n</ul>\n</li>\n<li>\n<p><strong>Differentiability (Real):</strong></p>\n<ul>\n<li>Using the power rule, the derivative of <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> is <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>f</mi><mi>′</mi></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>−</mo><mn>1</mn><mi>·</mi><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>=</mo><mo>−</mo><mn>1</mn><mo>/</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span>.</li>\n<li>This derivative <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>f</mi><mi>′</mi></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>−</mo><mn>1</mn><mo>/</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> exists for all <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow></math></span>.</li>\n<li>So, <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mi>x</mi></mrow></math></span> is once differentiable (in fact, infinitely differentiable) wherever it is defined.</li>\n</ul>\n</li>\n</ul>\n<p><strong>2. The Complex Extension: <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mi>z</mi></mrow></math></span></strong></p>\n<ul>\n<li>\n<p><strong>\"Shape\" (Mapping in the Complex Plane):</strong><br>\n Visualizing complex functions is different from real functions. We can't draw a simple 2D graph of input vs. output because both <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi></mrow></math></span> (input) and <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span> (output) are complex numbers, each having two components (real and imaginary). So, we'd need 4 dimensions. Instead, we often think about how <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span> <em>transforms</em> the complex plane:</p>\n</li>\n<li>\n<p><strong>Domain:</strong> <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mi>z</mi></mrow></math></span> is defined for all complex numbers <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi></mrow></math></span> except <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi><mo>=</mo><mn>0</mn></mrow></math></span> (the origin).</p>\n</li>\n<li>\n<p><strong>Representation:</strong> If <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi></mrow></math></span>, then <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mi>i</mi><mi>y</mi></mrow><mrow><mo stretchy=\"false\">(</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">(</mo><mi>x</mi><mo>−</mo><mi>i</mi><mi>y</mi><mo stretchy=\"false\">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mi>i</mi><mi>y</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mi>i</mi><mfrac><mrow><mi>y</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>.<br>\n So, the real part is <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>u</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mrow><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span> and the imaginary part is <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>v</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>−</mo><mfrac><mrow><mi>y</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>.</p>\n</li>\n<li>\n<p><strong>Geometric Interpretation (Inversion):</strong></p>\n<ul>\n<li>If we write <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi></mrow></math></span> in polar form as <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi><mo>=</mo><mi>r</mi><msup><mi>e</mi><mrow><mi>i</mi><mi>θ</mi></mrow></msup></mrow></math></span> (where <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>r</mi><mo>=</mo><mo stretchy=\"false\">|</mo><mi>z</mi><mo stretchy=\"false\">|</mo></mrow></math></span> is the modulus and <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>θ</mi><mo>=</mo><mi>\\arg</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span> is the argument), then <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>r</mi><msup><mi>e</mi><mrow><mi>i</mi><mi>θ</mi></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>r</mi></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>θ</mi></mrow></msup></mrow></math></span>.</li>\n<li>This means:<ul>\n<li>The modulus of <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span> is <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mn>1</mn><mo>/</mo><mo stretchy=\"false\">|</mo><mi>z</mi><mo stretchy=\"false\">|</mo></mrow></math></span>. Points inside the unit circle ( <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mo stretchy=\"false\">|</mo><mi>z</mi><mo stretchy=\"false\">|</mo><mo><</mo><mn>1</mn></mrow></math></span>, excluding 0) get mapped to points outside the unit circle (<span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mo stretchy=\"false\">|</mo><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">|</mo><mo>></mo><mn>1</mn></mrow></math></span>). Points outside get mapped inside. Points on the unit circle (<span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mo stretchy=\"false\">|</mo><mi>z</mi><mo stretchy=\"false\">|</mo><mo>=</mo><mn>1</mn></mrow></math></span>) stay on the unit circle.</li>\n<li>The argument of <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span> is <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mo>−</mo><mi>\\arg</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span>. This means the angle is reflected across the real axis.</li>\n</ul>\n</li>\n<li>This transformation is called an \"inversion\" with respect to the unit circle, followed by a reflection across the real axis.</li>\n</ul>\n</li>\n<li>\n<p><strong>Mapping Lines and Circles:</strong> <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mi>z</mi></mrow></math></span> maps lines and circles in the <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi></mrow></math></span>-plane to lines and circles in the <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span>-plane (where a line can be thought of as a circle of infinite radius). Specifically, circles passing through the origin in the <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi></mrow></math></span>-plane are mapped to lines in the <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span>-plane, and lines in the <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi></mrow></math></span>-plane are mapped to circles passing through the origin in the <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span>-plane (unless the line itself passes through the origin, in which case it's mapped to another line through the origin).</p>\n</li>\n<li>\n<p><strong>Differentiability (Complex):</strong></p>\n<ul>\n<li>To check for complex differentiability, we can use the definition of the derivative or the Cauchy-Riemann equations.</li>\n<li>Using the limit definition (similar to the real case, but <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>Δ</mi><mi>z</mi></mrow></math></span> is a complex increment):<br>\n<span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>F</mi><mi>′</mi></msup><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msub><mo>lim</mo><mrow><mi>Δ</mi><mi>z</mi><mo>→</mo><mn>0</mn></mrow></msub><mfrac><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo>+</mo><mi>Δ</mi><mi>z</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi>Δ</mi><mi>z</mi></mrow></mfrac><mo>=</mo><msub><mo>lim</mo><mrow><mi>Δ</mi><mi>z</mi><mo>→</mo><mn>0</mn></mrow></msub><mfrac><mrow><mn>1</mn><mo>/</mo><mo stretchy=\"false\">(</mo><mi>z</mi><mo>+</mo><mi>Δ</mi><mi>z</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mn>1</mn><mo>/</mo><mi>z</mi></mrow><mrow><mi>Δ</mi><mi>z</mi></mrow></mfrac></mrow></math></span><br>\n<span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mo>=</mo><msub><mo>lim</mo><mrow><mi>Δ</mi><mi>z</mi><mo>→</mo><mn>0</mn></mrow></msub><mfrac><mrow><mi>z</mi><mo>−</mo><mo stretchy=\"false\">(</mo><mi>z</mi><mo>+</mo><mi>Δ</mi><mi>z</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi>z</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo>+</mo><mi>Δ</mi><mi>z</mi><mo stretchy=\"false\">)</mo><mi>Δ</mi><mi>z</mi></mrow></mfrac><mo>=</mo><msub><mo>lim</mo><mrow><mi>Δ</mi><mi>z</mi><mo>→</mo><mn>0</mn></mrow></msub><mfrac><mrow><mo>−</mo><mi>Δ</mi><mi>z</mi></mrow><mrow><mi>z</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo>+</mo><mi>Δ</mi><mi>z</mi><mo stretchy=\"false\">)</mo><mi>Δ</mi><mi>z</mi></mrow></mfrac><mo>=</mo><msub><mo>lim</mo><mrow><mi>Δ</mi><mi>z</mi><mo>→</mo><mn>0</mn></mrow></msub><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mrow><mi>z</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo>+</mo><mi>Δ</mi><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></mfrac></mrow></math></span><br>\n As long as <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi><mo>≠</mo><mn>0</mn></mrow></math></span>, this limit is <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mo>−</mo><mn>1</mn><mo>/</mo><msup><mi>z</mi><mn>2</mn></msup></mrow></math></span>.</li>\n</ul>\n</li>\n<li>\n<p>So, <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mi>z</mi></mrow></math></span> is complex differentiable for all <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi><mo>≠</mo><mn>0</mn></mrow></math></span>, and its derivative is <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>F</mi><mi>′</mi></msup><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>−</mo><mn>1</mn><mo>/</mo><msup><mi>z</mi><mn>2</mn></msup></mrow></math></span>.</p>\n</li>\n<li>This means <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span> is holomorphic (analytic) on the domain <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>ℂ</mi><mi>⧵</mi><mo stretchy=\"false\">{</mo><mn>0</mn><mo stretchy=\"false\">}</mo></mrow></math></span> (the complex plane excluding the origin).</li>\n</ul>\n<p><strong>Connection:</strong><br>\nThe real function <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mi>x</mi></mrow></math></span> is indeed extendable to the complex function <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mi>z</mi></mrow></math></span>. When <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi></mrow></math></span> is a real number <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi></mrow></math></span> (i.e., <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mn>0</mn><mi>i</mi></mrow></math></span>), <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mi>x</mi></mrow></math></span>, which is <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span>. The complex function <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>F</mi><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></math></span> is differentiable (holomorphic) in its domain of definition (<span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>ℂ</mi><mi>⧵</mi><mo stretchy=\"false\">{</mo><mn>0</mn><mo stretchy=\"false\">}</mo></mrow></math></span>), which includes the domain of <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></math></span> (the real axis excluding <span><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span>).</p>","md":"Okay, let's break down $f(x) = 1/x$ and its extension $F(z) = 1/z$.\n\n**1. The Real Function: $f(x) = 1/x$**\n\n* **Initial Shape (Graph):**\n * This function describes a **hyperbola**.\n * It is defined for all real numbers $x$ except $x=0$.\n * For $x > 0$: As $x$ gets very small (approaches 0 from the positive side), $f(x)$ becomes very large and positive (approaches $+\\infty$). As $x$ gets very large, $f(x)$ gets very small and positive (approaches 0).\n * For $x < 0$: As $x$ gets very small in magnitude (approaches 0 from the negative side), $f(x)$ becomes very large and negative (approaches $-\\infty$). As $x$ gets very large in magnitude (but negative), $f(x)$ gets very small and negative (approaches 0).\n * The graph has two separate branches, one in the first quadrant and one in the third quadrant. The x-axis and y-axis are asymptotes.\n\n* **Differentiability (Real):**\n * Using the power rule, the derivative of $f(x) = x^{-1}$ is $f'(x) = -1 \\cdot x^{-2} = -1/x^2$.\n * This derivative $f'(x) = -1/x^2$ exists for all $x \\neq 0$.\n * So, $f(x) = 1/x$ is once differentiable (in fact, infinitely differentiable) wherever it is defined.\n\n**2. The Complex Extension: $F(z) = 1/z$**\n\n* **\"Shape\" (Mapping in the Complex Plane):**\n Visualizing complex functions is different from real functions. We can't draw a simple 2D graph of input vs. output because both $z$ (input) and $F(z)$ (output) are complex numbers, each having two components (real and imaginary). So, we'd need 4 dimensions. Instead, we often think about how $F(z)$ *transforms* the complex plane:\n * **Domain:** $F(z) = 1/z$ is defined for all complex numbers $z$ except $z=0$ (the origin).\n * **Representation:** If $z = x + iy$, then $F(z) = \\frac{1}{x+iy} = \\frac{x-iy}{(x+iy)(x-iy)} = \\frac{x-iy}{x^2+y^2} = \\frac{x}{x^2+y^2} - i\\frac{y}{x^2+y^2}$.\n So, the real part is $u(x,y) = \\frac{x}{x^2+y^2}$ and the imaginary part is $v(x,y) = -\\frac{y}{x^2+y^2}$.\n * **Geometric Interpretation (Inversion):**\n * If we write $z$ in polar form as $z = r e^{i\\theta}$ (where $r = |z|$ is the modulus and $\\theta = \\arg(z)$ is the argument), then $F(z) = \\frac{1}{r e^{i\\theta}} = \\frac{1}{r} e^{-i\\theta}$.\n * This means:\n * The modulus of $F(z)$ is $1/|z|$. Points inside the unit circle ( $|z|<1$, excluding 0) get mapped to points outside the unit circle ($|F(z)|>1$). Points outside get mapped inside. Points on the unit circle ($|z|=1$) stay on the unit circle.\n * The argument of $F(z)$ is $-\\arg(z)$. This means the angle is reflected across the real axis.\n * This transformation is called an \"inversion\" with respect to the unit circle, followed by a reflection across the real axis.\n * **Mapping Lines and Circles:** $F(z)=1/z$ maps lines and circles in the $z$-plane to lines and circles in the $F(z)$-plane (where a line can be thought of as a circle of infinite radius). Specifically, circles passing through the origin in the $z$-plane are mapped to lines in the $F(z)$-plane, and lines in the $z$-plane are mapped to circles passing through the origin in the $F(z)$-plane (unless the line itself passes through the origin, in which case it's mapped to another line through the origin).\n\n* **Differentiability (Complex):**\n * To check for complex differentiability, we can use the definition of the derivative or the Cauchy-Riemann equations.\n * Using the limit definition (similar to the real case, but $\\Delta z$ is a complex increment):\n $F'(z) = \\lim_{\\Delta z \\to 0} \\frac{F(z+\\Delta z) - F(z)}{\\Delta z} = \\lim_{\\Delta z \\to 0} \\frac{1/(z+\\Delta z) - 1/z}{\\Delta z}$\n $= \\lim_{\\Delta z \\to 0} \\frac{z - (z+\\Delta z)}{z(z+\\Delta z)\\Delta z} = \\lim_{\\Delta z \\to 0} \\frac{-\\Delta z}{z(z+\\Delta z)\\Delta z} = \\lim_{\\Delta z \\to 0} \\frac{-1}{z(z+\\Delta z)}$\n As long as $z \\neq 0$, this limit is $-1/z^2$.\n * So, $F(z) = 1/z$ is complex differentiable for all $z \\neq 0$, and its derivative is $F'(z) = -1/z^2$.\n * This means $F(z)$ is holomorphic (analytic) on the domain $\\mathbb{C} \\setminus \\{0\\}$ (the complex plane excluding the origin).\n\n**Connection:**\nThe real function $f(x)=1/x$ is indeed extendable to the complex function $F(z)=1/z$. When $z$ is a real number $x$ (i.e., $z=x+0i$), $F(x) = 1/x$, which is $f(x)$. The complex function $F(z)$ is differentiable (holomorphic) in its domain of definition ($\\mathbb{C} \\setminus \\{0\\}$), which includes the domain of $f(x)$ (the real axis excluding $x=0$).","profile":{"id":null,"name":null,"model":"gemini-2-5-pro","model_name":"Gemini 2.5 Pro","model_provider":"google","model_input_limit":1000000,"recommended":false,"model_info":"\n\n<div class=\"heading\">\n <div class=\"left\">\n \n \n \n <svg class=\"model-icon icon-xs\" data-model=\"gemini-2-5-pro\">\n <use href=\"#icon-google\"/>\n </svg>\n <h3>\n \n <span class=\"model-provider\">Google</span>\n \n <span class=\"model-name\">none</span>\n </h3>\n </div>\n\n \n <a href=\"/settings?p=custom_assistant&id=none\">\n <svg class=\"icon-sm edit-icon\" aria-hidden=\"true\">\n <use href=\"#icon-pencil\" />\n </svg>\n </a>\n \n</div>\n\n\n<hr />\n<ul>\n \n <li>\n <h4>\n Cost\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Relatively expensive\">\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --medium\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --medium\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --medium\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --medium\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-cost --medium\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Quality\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 4 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Speed\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 2 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Privacy\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 4 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li class=\"divider\" aria-hidden=\"true\"></li>\n <li>\n <h4>Last updated</h4>\n <span class=\"date\">\n \n \n 2025-06-17\n \n \n </span>\n </li>\n \n</ul>\n","model_provider_name":"Google","internet_access":true,"personalizations":true,"shortcut":null,"is_default_profile":false},"metadata":"<li>\n <span class=\"attribute\">Model</span>\n <span class=\"value\"><i class=\"icon-xs mr-2\"><svg><use href=\"#icon-google\"></use></svg></i>Gemini 2.5 Pro Preview</span>\n</li>\n<li>\n <span class=\"attribute\">Version</span>\n <span class=\"value\">models/gemini-2.5-pro-preview-05-06</span>\n</li>\n<li>\n <span class=\"attribute\">Speed (tok/s)</span>\n <span class=\"value\">48</span>\n</li>\n<li>\n <span class=\"attribute\">Tokens</span>\n <span class=\"value\">2,877</span>\n</li>\n<li>\n <span class=\"attribute\">Cost / Total ($)</span>\n <span class=\"value\">0.02 / 0.03</span>\n</li>\n<li>\n <span class=\"attribute\">End to end time (s)</span>\n <span class=\"value\">36.74</span>\n</li>\n\n","documents":[]}]