Fungrim home page

Fungrim entry: 2e4fbc

Symbol: IndefiniteIntegralEqual f(x)dx=g(x)+C\int f(x) \, dx = g(x) + \mathcal{C} Indefinite integral
IndefiniteIntegralEqual(f(x), g(x), x, c), rendered as f(x)dx=g(x)+C,x=c\int f(x) \, dx = g(x) + \mathcal{C}, x = c, expresses that g(x)g(x) is an antiderivative of f(x)f(x) at the point cc, or formally that g(c)=f(c)g'(c) = f(c). In other words, g(x)g(x) belongs to the equivalence class of antiderivatives of f(x)f(x) at the point cc. This is rendered as a statement of equality (with an arbitrary constant of integration) to follow the conventional notation for indefinite integrals.
This operator is ambiguous since the intended meaning could be a real derivative, a complex derivative, or some other form of derivative. It is better to use RealIndefiniteIntegralEqual or ComplexIndefiniteIntegralEqual.
The argument x defines a locally bound variable used in the expressions f(x)f(x) and g(x)g(x). If this operator is called more simply as IndefiniteIntegralEqual(f(x), g(x), x), the meaning is the same as IndefiniteIntegralEqual(f(x), g(x), x, x), where the x appearing in f(x)f(x) and g(x)g(x) is understood as a new dummy variable. This dummy variable is evaluated at the value x defined in the surrounding context only after the functions have been constructed.
Definitions:
Fungrim symbol Notation Short description
IndefiniteIntegralEqualf(x)dx=g(x)+C\int f(x) \, dx = g(x) + \mathcal{C} Indefinite integral
Derivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Derivative
RealIndefiniteIntegralEqualf(x)dx=g(x)+C\int f(x) \, dx = g(x) + \mathcal{C} Indefinite integral, real derivative
ComplexIndefiniteIntegralEqualf(x)dx=g(x)+C\int f(x) \, dx = g(x) + \mathcal{C} Indefinite integral, complex derivative
Source code for this entry:
Entry(ID("2e4fbc"),
    SymbolDefinition(IndefiniteIntegralEqual, IndefiniteIntegralEqual(f(x), g(x), x), "Indefinite integral"),
    Description(SourceForm(IndefiniteIntegralEqual(f(x), g(x), x, c)), ", rendered as ", IndefiniteIntegralEqual(f(x), g(x), x, c), ", expresses that", g(x), "is an antiderivative of", f(x), "at the point", c, ", or formally that", Equal(Derivative(g(x), For(x, c)), f(c)), ".", "In other words,", g(x), "belongs to the equivalence class of antiderivatives of", f(x), "at the point", c, ". This is rendered as a statement of equality (with an arbitrary constant of integration) to follow the conventional notation for indefinite integrals."),
    Description("This operator is ambiguous since the intended meaning could be a real derivative, a complex derivative, or some other form of derivative.", "It is better to use", SourceForm(RealIndefiniteIntegralEqual), "or", SourceForm(ComplexIndefiniteIntegralEqual), "."),
    Description("The argument", SourceForm(x), "defines a locally bound variable used in the expressions", f(x), "and", g(x), ". ", "If this operator is called more simply as", SourceForm(IndefiniteIntegralEqual(f(x), g(x), x)), ", ", "the meaning is the same as", SourceForm(IndefiniteIntegralEqual(f(x), g(x), x, x)), ", where the", SourceForm(x), "appearing in", f(x), "and", g(x), "is understood as a new dummy variable. This dummy variable is evaluated at the value", SourceForm(x), "defined in the surrounding context only after the functions have been constructed."))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-08-27 09:56:25.682319 UTC