Branch point

Branch point

In the mathematical field of complex analysis, a branch point may be informally thought of as a point "z"0 at which a "multi-valued function" (usually referred to as a "multifunction" in the context of complex analysis) changes values when one winds once around "z"0.

Examples:

* 0 is a branch point of the square root function. Suppose "w" = √"z", and "z" starts at 4 and moves along a circle of radius 4 in the complex plane centered at 0. The dependent variable "w" changes while depending on "z" in a continuous manner. When "z" has made one full circle, going from 4 back to 4 again, "w" will have made one half-circle, going from the positive square root of 4, i.e., from 2, to the negative square root of 4, i.e., −2.

* 0 is also a branch point of the natural logarithm. Since "e"0 is the same as "e"2π"i", both 0 and 2π"i" are among the multiple values of Log(1). As "z" moves along a circle of radius 1 centered at 0, "w" = Log("z") goes from 0 to 2π"i".

* In trigonometry, since tan(π/4) and tan (5π/4) are both equal to 1, the two numbers π/4 and 5π/4 are among the multiple values of arctan(1). The imaginary units "i" and −"i" are branch points of the arctangent function (arctan("z") = (1/2"i")log("i" − "z")/("i" + "z")). This may be seen by observing that the derivative ("d"/"dz") arctan("z") = 1/(1 + "z"2) has simple poles at those two points, since the denominator is zero at those points.

* If the derivative "f" ' of a function "f " has a simple pole at a point "a", then "f" has a branch point at "a". (The converse is false, since the square-root function is a counterexample.)

In order to work with single-valued functions, it is customary to construct branch cuts in the complex plane, namely arcs out of branch points in the complement of which there is a well-defined branch of the function in question. An example for

:F(z) = sqrt{z} sqrt{1-z},

is to make a branch cut along the interval [0, 1] on the real axis, connecting the two branch points of the function. The same idea can be applied to the function √"z"; but in that case one has to perceive that the "point at infinity" is the appropriate 'other' branch point to connect to from 0, for example along the whole negative real axis. See also principal branch.

The branch cut device may appear arbitrary (it is); but it is very useful, for example in the theory of special functions. An invariant explanation of the branch phenomenon is developed in Riemann surface theory (of which it is historically the origin), and more generally in the ramification and monodromy theory of algebraic functions and differential equations.


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • branch point — šakojimosi taškas statusas T sritis automatika atitikmenys: angl. branch point; branching point; branchpoint vok. Verzweigungspunkt, m; Verzweigungsstelle, f rus. точка ветвления, f pranc. bifurcation, f; point de branchement, m …   Automatikos terminų žodynas

  • branch point — 1. Elect. a point in an electric network at which three or more conductors meet. 2. Math. a point such that analytic continuation of a given function of a complex variable in a small neighborhood of the point produces a different functional value …   Universalium

  • branch point — 1. Elect. a point in an electric network at which three or more conductors meet. 2. Math. a point such that analytic continuation of a given function of a complex variable in a small neighborhood of the point produces a different functional value …   Useful english dictionary

  • Skirmish at Many Branch Point — Part of Falklands War Date 10 June 1982 Location North of Port Howard, West Falkland …   Wikipedia

  • Engagement at Many Branch Point — Infobox Military Conflict conflict=Commandos Clash at Many Branch Point partof=Falklands War campaign= caption= date= 10 June 1982 place=Port Howard, West Falkland result=Argentine victory combatant1=Flagicon|Argentina Argentina… …   Wikipedia

  • Branch (disambiguation) — A branch is a part of a woody plant.Branch or branching may also refer to:Computing* Branch (computer science), a point in a computer program where program flow may change depending on a condition. **Branch predictor, the part of a processor that …   Wikipedia

  • point de branchement — šakojimosi taškas statusas T sritis automatika atitikmenys: angl. branch point; branching point; branchpoint vok. Verzweigungspunkt, m; Verzweigungsstelle, f rus. точка ветвления, f pranc. bifurcation, f; point de branchement, m …   Automatikos terminų žodynas

  • Branch and bound — (BB) is a general algorithm for finding optimal solutions of various optimization problems, especially in discrete and combinatorial optimization. It consists of a systematic enumeration of all candidate solutions, where large subsets of… …   Wikipedia

  • Branch and cut — (sometimes written as branch and cut ) is a method of combinatorial optimization for solving integer linear programs, that is, linear programming problems where some or all the unknowns are restricted to integer values. The method is a hybrid of… …   Wikipedia

  • point — [point] n. [OFr, dot, prick < L punctum, dot, neut. of punctus, pp. of pungere, to prick (< IE base * peuĝ , *peuk̑, to prick, jab > Ger fichte, spruce tree, L pugil, boxer, pugnus, fist); also < OFr pointe, sharp end < ML puncta… …   English World dictionary

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”