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In fact, holomorphy is characterized by having an antiderivative locally, that is, g is holomorphic if for every z in its domain, there is some neighborhood U of z such that g has an antiderivative on U. Furthermore, holomorphy is a necessary condition for a function to have an antiderivative, since the derivative of any holomorphic function is ...
The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a continuous function f is a differentiable function F whose derivative is equal to the original function f.
On any simply connected neighborhood U in C − {0}, 1/z has an antiderivative defined by L(z) = ln(r) + iθ, where z = re iθ. Because of the ambiguity of θ up to the addition of any integer multiple of 2 π, any continuous choice of θ on U will suffice to define an antiderivative of 1/z on U.
In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards."
The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity.
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x) = 0.
Nevertheless, it can be shown that this theorem holds even if or is not differentiable: [3] [4] it suffices, for example, to use the Stieltjes integral in the previous argument. On the other hand, even though general monotonic functions are differentiable almost everywhere, the proof of the general formula does not follow, unless f − 1 ...
In calculus, the constant of integration, often denoted by (or ), is a constant term added to an antiderivative of a function () to indicate that the indefinite integral of () (i.e., the set of all antiderivatives of ()), on a connected domain, is only defined up to an additive constant.