Analytization trick
Encyclopedia
The analytization trick is a heuristic
often applied by physicist
s.
Suppose we have a function
f of a complex variable z which is not analytic
, but happens to be differentiable with respect to its real
and imaginary
components separately. Differentiating f with respect to z is out of the question, but it turns out if
for some analytic function
g of two complex variables, we can pretend f is g (physicists do this sort of thing all the time) and work with
and
instead. Physicists write these as
and
and give some handwaving
explanation as to why
and z may be treated as if they are "independent" when they really are not.
Note that if g exists, it is unique (due to the theorem about the uniqueness of analytic continuation
s), at least if we ignore complications like branch cuts and so on.
Conceptually, whenever this trick is used, it probably means on a physical level that the variable z that they are working with "really" has a real structure and physicists are merely pigeonholing it into a complex variable.
Actually, it's not even necessary for there to be an analytic g. It's enough for f to be differentiable with respect to its real
and imaginary
components (or n times differentiable, as the case may be). In that case,
has to be treated purely formally.
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...
often applied by physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
s.
Suppose we have a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f of a complex variable z which is not analytic
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
, but happens to be differentiable with respect to its real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
and imaginary
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...
components separately. Differentiating f with respect to z is out of the question, but it turns out if
for some analytic function
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
g of two complex variables, we can pretend f is g (physicists do this sort of thing all the time) and work with
and
instead. Physicists write these as
and
and give some handwaving
Handwaving
Handwaving is a pejorative label applied to the action of displaying the appearance of doing something, when actually doing little, or nothing. For example, it is applied to debate techniques that involve logical fallacies. It is also used in working situations where productive work is expected,...
explanation as to why
and z may be treated as if they are "independent" when they really are not.Note that if g exists, it is unique (due to the theorem about the uniqueness of analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
s), at least if we ignore complications like branch cuts and so on.
Conceptually, whenever this trick is used, it probably means on a physical level that the variable z that they are working with "really" has a real structure and physicists are merely pigeonholing it into a complex variable.
Actually, it's not even necessary for there to be an analytic g. It's enough for f to be differentiable with respect to its real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
and imaginary
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...
components (or n times differentiable, as the case may be). In that case,
has to be treated purely formally.