Path integral
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- This article is about "path integrals" in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman.
In mathematics, a path integral (also known as a line integral) is an integral where the function to be integrated is evaluated along a path or curve. Various different path integrals are in use. In the case of a closed path it is also called a contour integral.
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Complex analysis
The path integral is a fundamental tool in complex analysis. Suppose U is an open subset of C, γ : [a, b] → U is a rectifiable curve and f : U → C is a function. Then the path integral
- <math>\int_\gamma f(z)\,dz<math>
may be defined by subdividing the interval [a, b] into a = t0 < t1 < ... < tn = b and considering the expression
- <math>\sum_{1 \le k \le n} f\left( \;\gamma(t_k)\;\right) \left[ \; \gamma(t_k) - \gamma(t_{k-1}) \; \right].<math>
The integral is then the limit of this sum as the lengths of the subdivision intervals approach zero.
If γ is a continuously differentiable curve, the path integral can be evaluated as an integral of a function of a real variable:
- <math>\int_\gamma f(z)\,dz
=\int_a^b f(\gamma(t))\,\gamma\,'(t)\,dt.<math>
When γ is a closed curve, that is, its initial and final points coincide, the notation
- <math>\oint_\gamma f(z)\,dz<math>
is often used for the path integral of f along γ.
Important statements about path integrals are the Cauchy integral theorem and Cauchy's integral formula.
Because of the residue theorem, one can often use contour integrals in the complex plane to find integrals of real-valued functions of a real variable (see residue theorem for an example).
Example
Consider the function f(z)=1/z, and let the contour C be the unit circle about 0, which can be parametrized by eit, with t in [0, 2π]. Substituting, we find
- <math>\oint_C f(z)\,dz = \int_0^{2\pi} {1\over e^{it}} ie^{it}\,dt = i\int_0^{2\pi} e^{-it}e^{it}\,dt<math>
- <math>=i\int_0^{2\pi}\,dt = i(2\pi-0)=2\pi i<math>
which can be also verified by the Cauchy integral formula.
Vector calculus
In qualitative terms, a path integral in vector calculus can be thought of as a measure of the effect of a given vector field along a given curve.
Definition
For some scalar field f : Rn → R, the path (or line) integral on a curve C, parametrized as r(t) with t ∈ [a, b], is defined by
- <math>\int_C f\ ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)| dt.<math>
Similarly, for a vector field F : Rn → Rn, the path integral on a curve C, parametrized as r(t) with t ∈ [a, b], is defined by
- <math>\int_C \mathbf{F}(\mathbf{x})\cdot\,d\mathbf{x} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt.<math>
Path independence
If a vector field F is the gradient of a scalar field G, that is,
- <math>\nabla G = \mathbf{F},<math>
then the derivative of the composition of G and r(t) is
- <math>\frac{dG(\mathbf{r}(t))}{dt} = \nabla G(\mathbf{r}(t)) \cdot \mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)<math>
which happens to be the integrand for the path integral of F on r(t). It follows that, given a path C , then
- <math>\int_C \mathbf{F}(\mathbf{x})\cdot\,d\mathbf{x} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt = \int_a^b \frac{dG(\mathbf{r}(t))}{dt}\,dt = G(\mathbf{r}(b)) - G(\mathbf{r}(a)).<math>
In words, the integral of F over C depends solely on the values of the points r(b) and r(a) and is thus independent of the path between them.
For this reason, a vector field which is the gradient of a scalar field is called path independent.
Applications
The path integral has many uses in physics. For example, the work done on a particle traveling on a curve C inside a force field represented as a vector field F is the path integral of F on C.
Quantum mechanics
The "path integral formulation" of quantum mechanics actually refers not to path integrals in this sense but to functional integrals, that is, integrals over a space of paths, of a function of a possible path. However, path integrals in the sense of this article are important in quantum mechanics; for example, complex contour integration is often used in evaluating probability amplitudes in quantum scattering theory.