Fungrim entry: fb2790

Symbol: ComplexIndefiniteIntegralEqual —

\int f(x) \, dx = g(x) + \mathcal{C}

— Indefinite integral, complex derivative

ComplexIndefiniteIntegralEqual(f(x), g(x), x, c), rendered as

\int f(x) \, dx = g(x) + \mathcal{C}, x = c

, expresses that

g(x)

is an antiderivative of

f(x)

at the point

c

, or formally that

g'(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.

The argument x defines a locally bound variable used in the expressions

f(x)

and

g(x)

. If this operator is called more simply as ComplexIndefiniteIntegralEqual(f(x), g(x), x), the meaning is the same as ComplexIndefiniteIntegralEqual(f(x), g(x), x, x), where the x appearing in

f(x)

and

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
`ComplexIndefiniteIntegralEqual`	$\int f(x) \, dx = g(x) + \mathcal{C}$	Indefinite integral, complex derivative
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative

Source code for this entry:

Entry(ID("fb2790"),
    SymbolDefinition(ComplexIndefiniteIntegralEqual, ComplexIndefiniteIntegralEqual(f(x), g(x), x), "Indefinite integral, complex derivative"),
    Description(SourceForm(ComplexIndefiniteIntegralEqual(f(x), g(x), x, c)), ", rendered as ", ComplexIndefiniteIntegralEqual(f(x), g(x), x, c), ", expresses that", g(x), "is an antiderivative of", f(x), "at the point", c, ", or formally that", Equal(ComplexDerivative(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("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(ComplexIndefiniteIntegralEqual(f(x), g(x), x)), ", ", "the meaning is the same as", SourceForm(ComplexIndefiniteIntegralEqual(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

Operators