An integral representation expresses a function through an integral of specified data and an integral kernel. Green-function representations solve differential equations this way, while the Bromwich inversion formula reconstructs a function from its Laplace transform.
Lebesgue integration constructs an integral from the measure of level sets or from approximation by simple functions. It is linear on integrable functions and is stable under the principal convergence theorems of measure theory.
If and nonnegative measurable functions satisfy for all , then their Lebesgue integrals obeyThe quantile derivative identity and monotone substitution inequality prove the one-dimensional case; Tonelli theorem extends it by induction. It implies the Brunn–Minkowski inequality.
If , then for every there is such that whenever the measure of is below . In particular, integrals over a sequence of shrinking measurable sets tend to zero.
A time integral accumulates an integrand over a time coordinate. In perturbation theory it commonly sums the effect of an interaction over all possible vertex times.
A double integral integrates a function over a two-dimensional region. A change of variables formula can replace the region and area element using a Jacobian determinant.
Any two antiderivatives of the same function on a connected interval differ by an additive constant.
The Dawson function obeys by differentiation. Its large-real-argument asymptotic expansion begins . This inverse-power behavior contrasts with the separate exponentially growing factors in its definition.
Define and . They satisfy and . The first is an even function, the second an odd function, and their real tails are respectively and . Their derivatives have inverse-linear tails, explaining the logarithmic matching constants in a singular perturbation.
The complementary error function avoids subtracting a quantity close to one when describing a small Gaussian integral tail. For real , ; analytic continuation defines it for complex arguments.
The Faddeeva function is an entire function built from the complementary error function. Its integral representation for is , as normalized in NIST's integral representation. Its reflection identity is , directly from the odd function property of the error function.
For a pole off the real axis, substitution into the Faddeeva function integral gives when , and when . The difference of boundary values is the residue theorem jump . In particular, for a lower-half-plane pole, , which packages a crossed residue and a nearby saddle into one smooth expression.
Euler's number is the value of the exponential function, approximately . It is the base of the natural logarithm. Its factorial-series tail gives an elementary proof of the irrationality of e.
If Euler's number were , choose and multiply its factorial-series tail by . This produces an integer, but the tail is positive and smaller than . The contradiction proves that the number is an irrational number. The argument illustrates how a rapidly convergent rational series can force a hypothetical rational remainder into an impossible integer interval.
For real , the complex exponential function has . Since the trigonometric factor lies on the unit circle,Thus the real part of the exponent controls its modulus, while the imaginary part controls its direction on the complex plane.
The kernel of the complex exponential function givesThe same period appears as the translation generated by the monodromy of the complex inverse sine.
For integers , rationality of is equivalent to equality of some positive integer powers . Exponentiation proves one direction and taking logarithms proves the other. Thus multiplicatively independent integers give an irrational number as their logarithm ratio. For distinct prime bases and arguments, unique factorization supplies that independence.
The logarithm is strictly concave on positive real numbers, as its second derivative is negative. Thus the Jensen inequality gives for a probability measure and an integrable positive , whenever the left side is defined. It also controls the negative logarithmic part of convex combinations of strictly positive functions.
The Cauchy principal value integral is the logarithmic integral. The offset version differs by a constant and is often used with the prime-counting function. It has .
For positive-definite Hermitian operators,The same statement on supports follows by regularizing with a positive multiple of the identity and taking a limit.
A Gaussian function has the form with . Its integral and moments reduce by translation and scaling to the Gaussian integral.
For fixed , and , complete the square in the exponent. A translated Gaussian function summed on a fixed-spaced lattice has mass uniformly in the translate, by comparison with its integral and its maximum. This keeps a square-root logarithm, rather than the full number of dyadic intervals, when summing exponentially damped exponential sum bounds.
Hyperbolic functions are combinations of the exponential function analogous to trigonometric functions. The hyperbolic cosine and hyperbolic sine satisfy and parametrize a unit hyperbola.
The hyperbolic cosine is .
The hyperbolic sine is .
The hyperbolic-sine infinite product gives, by taking its logarithmic derivative,The identity holds away from the poles, with locally convergent sums. For real , it makes positive and strictly decreasing. It is useful for Hartmann flow flux monotonicity and small-argument expansions of hyperbolic functions.
The hyperbolic secant is
If , then is absolutely continuous and satisfies almost everywhere. Conversely, every absolutely continuous function has this form.
For real functions continuous on and differentiable on , there exists withApply Rolle's theorem to a linear combination of the functions with equal endpoint values. If never vanishes, and the equality can be divided into a ratio. This form directly proves the zero-over-zero endpoint case of L'Hôpital's rule, without continuity of the derivatives.
If is continuous at an interval endpoint, differentiable in its interior, and has a finite one-sided derivative there, then its secant slope and tangent slope are arbitrarily close at some interior point. Apply the mean value theorem on a short endpoint interval; both the secant at its far endpoint and the secant at the resulting intermediate point approach the same one-sided derivative. No continuity of the derivative is required.
If a continuous real function is differentiable between two distinct points at which it has the same value, its derivative vanishes at some point between them.
If a nonzero real polynomial has roots in an open real interval, counted with multiplicity of a root, its derivative has at least roots in that interval, also counted with multiplicity, provided . Each root of multiplicity becomes a root of the derivative of multiplicity . The Rolle theorem supplies one further root between each pair of distinct consecutive roots. Thus distinct roots of multiplicities give at least derivative roots.
The Taylor theorem expresses a sufficiently differentiable function as a finite Taylor polynomial plus a remainder.
For a continuously differentiable vector-valued function whose domain contains the segment from to , apply the fundamental theorem of calculus componentwise to . The resulting integral matrix is valid even when there is no common intermediate point giving a vector mean-value formula. For a score function, it integrates the Hessian matrix along the segment to a candidate estimate.
A Taylor expansion about a point consists of successive derivative terms there, together with either a finite remainder or an infinite Taylor series when that series represents the function.
A Peano zero through order means that the function admits the zero Taylor polynomial with remainder at . This does not assert the existence of ordinary iterated derivatives on a neighborhood. For example, multiplying by a bounded continuous function with dense points of nondifferentiability preserves this estimate while destroying neighborhood differentiability.
A parabolic approximation retains terms through second order in a local Taylor expansion. Near the closest point of a circle of radius and its tangent line, the separation is .
The Taylor remainder is the difference between a function and a finite Taylor polynomial. Bounds on higher derivatives or their Hölder continuity control this error.
If is locally absolutely continuous, then its Taylor remainder can be written as an integral involving the weak derivative . This form permits norm bounds when the highest derivative is only integrable.
A one-sided limit restricts the argument to values above or below the limiting point. It is denoted from above and from below.
Bounded continuous functions on a topological space form a Banach space with the supremum norm. A uniform limit is continuous and bounded, proving completeness. This space differs from the bounded uniformly continuous functions: free-transport composition is jointly continuous in time and position, but need not be continuous in the global sup norm as time varies.
A real or extended-real function is upper semicontinuous when every strict sublevel set is open. Equivalently its negative is lower semicontinuous. An arbitrary infimum of continuous functions is upper semicontinuous, even when the infimum is over an uncountable family; its strict sublevel sets are unions of the corresponding open sublevel sets.
If near and as , then by the squeeze theorem, even if has no limit. Defining the product to be zero at gives continuity there. For example, extends continuously at zero; the same oscillation near a nonzero multiple of has nonvanishing amplitude and does not have a continuous zero extension.
A function is lower semicontinuous when its strict superlevel sets are open, equivalently when its sublevel sets are closed. In Euclidean spaces this is the sequential lower semicontinuity condition whenever . This closedness property is essential to existence of proximal minimizers.
A sublevel set consists of points whose objective value is at most a chosen threshold. Coercivity makes finite sublevel sets bounded, while sequential lower semicontinuity makes them sequentially closed sets in the selected topology.
On a metric space, a functional is sequentially lower semicontinuous exactly when every sublevel set is closed.
A function is locally constant when every point has a neighbourhood on which the function is constant.
If is continuous and lies between and , then some satisfies .
A continuous injective real-valued function on a real interval is strictly monotone. A value outside the interval between two endpoint values would, by the intermediate value theorem, create a repeated value. Applying this observation to subintervals gives consistent strict ordering. If the left endpoint value is smaller than the right endpoint value, the function is strictly increasing; otherwise it is strictly decreasing.
If is continuous and , then for every positive integer there is with . For , the continuous difference satisfies . Either one sampled difference is zero or two have opposite signs; the intermediate value theorem then supplies a zero between them. For , the endpoint chord suffices.
If a continuous real-valued function on a real or complex vector space has opposite signs at two points, restricting it to the line segment between them and applying the intermediate value theorem gives a zero.
Every continuous injective function from a real interval to the real line is strictly monotone. A continuous bijection therefore has a continuous inverse.
If is continuous at and is continuous at , then is continuous at .
The continuity set of a function is the set of points where it is continuous. For a real function , define its oscillation at byThe sets are open, and is continuous at exactly when . Hence its continuity set is the Borel set
A càdlàg function is right-continuous and has a finite left limit at every positive time. The name abbreviates the French phrase /continue à droite, limites à gauche/.
A nonnegative uniformly continuous function on with finite integral tends to zero. Otherwise uniform continuity gives disjoint intervals of a fixed width on which the function exceeds a fixed positive value, contradicting integrability. This supplies a useful dissipation argument for Lyapunov functions with merely continuous vector fields.
The derivative is the limit of the difference quotient and gives the best linear approximation to local change.
The tangent line to a differentiable function at is its affine first-order approximation . For a differentiable concave function, it is an upper bound throughout a convex interval: the secant-slope inequality and passage to the derivative give . This upper bound constructs valid envelopes in adaptive rejection sampling.
Automatic differentiation evaluates derivatives of a computational graph using the chain rule. Reverse mode is backpropagation and is useful for reparameterization gradients through simulated trajectories.
Differentiation computes the derivative of a function. A difference quotient approximates a derivative using finite increments; its limit is the derivative when the function is differentiable.
The derivative of the first derivative. For a twice differentiable function, its sign governs local convexity and its magnitude controls the second-order Taylor remainder.
A difference quotient divides a finite change in a function by the corresponding change in its argument. Its limit, when it exists, is the derivative.
With , a change of variables givesThe compact support of and its translates must remain inside the integration domain, or one must integrate over the whole space with convergent integrals. In particular : the denominator's sign matters. This is the difference quotient analogue of integration by parts.
A total derivative accounts for every dependence of a quantity on the variable of differentiation. Its integral contributes only boundary terms.
For an indeterminate quotient whose numerator and denominator both tend to zero or both diverge, suitable differentiability and denominator conditions allow its limit to be computed from the limit of the quotient of their derivatives.
Every derivative has the intermediate-value property, even when the derivative is discontinuous.
The higher-order product rule is
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Calculus is a branch of mathematics that deals with the study of change and motion. It focuses on concepts such as limits, derivatives, integrals, and infinite series. Calculus is primarily divided into two main branches: 1. **Differential Calculus**: This branch focuses on the concept of the derivative, which represents the rate of change of a function with respect to a variable.
