FIG I.
V = const. going over into the orthogonal family, whose asymptotes are these axes.
If u and v have continuous first partial derivatives and if their Jacobian au au Ox ay av av ax ay does not vanish. then at least for a restricted region Se about a point (5., y.), the equations (2) will always define a one-to-one map of Se on a region E. including the point (th, v,). This map will represent approximately (i.e., to in finitesimals of higher order than the first) a pro jection in other words strain of the imme diate neighborhood of (xe, ye) on that of (u., ve), a small circle about (xeye) going approximately into an ellipse about (u., v.). If, however, so and v are conjugate functions, the ellipse be comes a circle, and the angle under which two curves intersect in the (z,y)-plane is preserved in the (a, v)-plane. Thus the shapes of small figures are but slightly distorted. Such a map is called conformal or isogonal. The elements of corresponding infinitesimal arcs in the two planes are connected by the relation dS=Mds, where M is a positive function of x and y, and does not depend on ds, dS. The Jacobian here has the value j.= (-814\ t_av \axi \axi Referring to the developments of § 3 we see that the theorem just stated is coextensive with the following: Theorem: If r.tf (z) is single-valued and analytic in the region S, then this functiotr maps the neighborhood of any point m of S, in which f (z.) *0, in a one-to-one manner and conformally on the neighborhood of the point wo=f (4).
The problem of conformal mapping first studied was that of making a map of the earth such that the shapes of small regions should be but slightly distorted. The two principal solutions that presented themselves were (a) Ptolemy's Projection, known in mathematics as stereographic projection, which consists in pass ing a variable ray through a fixed point of the sphere, which we will think of as the north pole. Let P and Q be respectively the variable Intersections of this ray with the sphere and with the plane of the equator, or a parallel plane, taken as the plane of the projection.
Then Q is the projection of P. (b) Mercator's Chart.
By a simply connected region is meant a region bounded by a single curve. Any simply connected region can he mapped in a one-to-one manner and conformally on the interior of a circle, and hence any two such regions can be mapped conformally on each other (Riemann, 1851) 5. ;The Elementary Functions. (a) The Exponential Function. For real values of z the function ex can be expanded by Tayloes Theorem into the series ez x 2! 3! Substituting formally for x a pure imaginary value and operating with the infinite series as if it were a polynomial, we get 1 Y2 Ys 2! 4! 6! ("" 3! S! 7! 7 =cos y+i sin y.
This formal work suggests a definition of ex for a pure imaginary value of the argument, namely: esix-tos y-Fi sin y, (3) and it is this definition that we shall adopt, setting it at the satne time more generally es= ex + ez (cos y sio y).
The function of ex thus obtained is a true generalization of ex, for it is single-valued and analytic for all values of .7., reduces to ex when z is real, and satisfies the same functional rela tions as ez : des ezi+ is, es.
ds (b) The Trigonometric Functions. When x is real, we have from (3) ea_ ;-21i sin COS 21 2i 2 The right-hand sides of these equations have a meaning for complex values of the argument. By means of them we define the functions sin z, cos c: est: e--11 eii Sin COS z=- 2i 2 These generalized functions are single valued and analytic for all values of z and sat isfy the same functional relations as sin x, cos x; e.g., sin (zi-l-c.)--sin zi cos ze-Fcos zi sin zi. The tan z is defined as sin z/cos z, etc.
(c) The Inverse Functions. The function log z shall be defined as the inverse of the ex ponential function: evx=z, wlog Setting w=-.-u +vi z--r (cos 9$ i sin 99, we have eu(cos v i sin v) =----r(cos + i sin 0). Hence log r, v=0 -I- 2kir, ands=log 1+ (0 ± 2100i Thus it appears that the logarithm is no longer single-valued when complex numbers are ad mitted to consideration. For example, log 2=.693...+2/riri.