Consistent transformations
1) consistent Transformation :
It is a transformation which possesses the property to conserve the angles (the angle between 2 regular curves whichever intersecting themselves in a point z is equal in module and sense of spin to the angle between their images) and to expand in constant way (infinitely small segments is transformed in similar segments).
In other words an application is said consistent if biunivoca and having the property of costanza of the expansions and costanza of the angles.
2) necessary and sufficient Condition for the conformity:
An application is consistent > the function of a variable complex is univoca, analytics and its derivative is various from 0 in every point of the dominion.
3) consistent Application of second species :
Draft of an application which conserve the absolute values of the angles forms from 2 curves and their images but not of it conserve to you the sense. Draft of applications produced from functions of a variable complex that are complex functions conjugated of functions analytics with never null derivative.
4) Effect of one linear application :
It supplies one similar expansion of the plan z and one translation of the origin of the coordinates, pu² therefore to be used in order to construct consistent applications of similar figures.
5) When the function 1/z applies one circumference on one straight :
When the circumference passes for the origin.
6) Effect of the function power :
It applies a circular field on all the cut plan, is a consistent application for all the points to exception of the points of frontier z = 0 e z = ¥ in which the derivative before cancels itself.
7) Effect of one exponential application :
It biunivocamente applies to every strip parallel to the real axis on all the cut plan, possesses derived various from 0 and therefore exponential application is one.
8)Principi it generates them of the consistent applications :
to) Correspondence biunivoca between the dominion and the image
b) It is necessary to only control that the tried function applies the frontier of the dominion on the frontier of the image.
c) Principle of symmetry
9) Principle of correspondence of the frontiers :
A dominion limited from a contour is G g on which a function is defined analytics univoca f(z) continues that it applies in way biunivoco the contour g on a curve G sluice of the complex plan w if in such application of curves sluices conserve the spin back, the function f(z) gives a consistent application of the G dominion on the limited inner dominion from the G contour.
It is demonstrated applying to the theorem of the
argument to the functions members of the women's army auxiliary corps
F1(z) = f(z) - w1 e F2(z) = f(z) - w2 where w1 is
an inner point to G and w2 are instead an external
point obtain 
and also 
2 that wants to say that 2 is not never
had f(z)=w beingw a generic external point to the
dominion and that the application is biunivoca in fact exists a
solo zero of f(z) = w1 .
10) If f(z) it is analytics in the dominion except that for a singolarità point, then it applies the inside of the edge of the dominion on the outside of the circuit that is image of the edge of the dominion.
11) Theorem of Riemann :
Every simply connected G dominion of the complex plan z, whose frontier consists more than a point, can be applied in consistent way on the inside of the unitary circle |w|< 1 of the plan w.
12) the function f(z) that it produces a consistent application of a dominion given sempliecemente connected G on the unitary circle so that z0 go in the origin and argf ' (z0) is one constant is defined in only way:
13) Function rations to delineate them :
The shape is a having 
function,
realizes an application in compliance with pact that takes place the
iniettività, in particular given 2 points z1 and z2 they
have various image if 
as it is obtained placing 
.
Equivalent can be written 
con 
in
order to obtain which enough to collect the coefficients of the z is
to the numerator that to the denominator.
14) symmetrical Points respect to one circumference :
The points P and P' are symmetrical regarding circumference C if giacciono on one same beam passing for the center of the circumference and the product of their distances from the center he is equal to the square of the beam of the circumference.
15) a function rations to delineate them is defined in only way if correspondence between 3 is assigned one distinguished points of the plan z and 3 distinguished points of the plan w.
16) Property of the function rations to delineate them :
to) it transforms the circumferences of the plan z in circumferences of the plan w
b) the symmetrical points respect to every circumference are transformed in symmetrical points regarding the image of this circumference.
17) biangular Figure :
Draft of a flat figure constituted from the intersection of is arched of 2 circumferences of beams in a generalized manner distinguished.
18) Function of Zukovsky :
It is the 
function, deriving it it
is looked at that he is consistent ovunque unless in the points 1
e -1.
It transforms circumferences centered in the outgoing origin in ellipses and beams from the origin in hyperbolas.