. The Americana : a universal reference library, comprising the arts and sciences, literature, history, biography, geography, commerce, etc. of the world . Let w=ex(cos y+i sin y). Then du dv — = ex cos y=-z~,dx dy du dv — --e*smy--rx. Hence w has a derivative, and furthermoredw dz = M/. A function which is single-valued throughouta region S and has a continuous derivative inS is said to be analytic in S. The terms holo-tnorphic, monogenic, and synectic are also some-times used in this sense. 4. Conformal Mapping.—Given any two func-tions u = (x,y), v-tp(x,y), (2) we may interpret them geometr
. The Americana : a universal reference library, comprising the arts and sciences, literature, history, biography, geography, commerce, etc. of the world . Let w=ex(cos y+i sin y). Then du dv — = ex cos y=-z~,dx dy du dv — --e*smy--rx. Hence w has a derivative, and furthermoredw dz = M/. A function which is single-valued throughouta region S and has a continuous derivative inS is said to be analytic in S. The terms holo-tnorphic, monogenic, and synectic are also some-times used in this sense. 4. Conformal Mapping.—Given any two func-tions u = (x,y), v-tp(x,y), (2) we may interpret them geometrically as trans-forming the points of one plane into the points(u, v) of a second plane. Thus a region S ofthe first plane will be mapped, if certain furtherconditions are fulfilled, in a one-to-one mannerand continuously on a region I of the (w, v)-plane, For example, let u=xi— y, v = 2xy. Here the first quadrant of the (x, ;y) on the upper half of the («, v)-plane,the family of lines u= const, going over into the COMPLEX VARIABLE family of equilateral hyperbolas whose axeslie in the coordinate axes, and the family of. Fig. du du ~dx by dv dv Tx dy lines v = const, going over into the orthogonalfamily, whose asymptotes are these axes. If u and v have continuous first partial deriva-tives and if their Jacobian / does not vanish, then, at least for a restrictedregion S0 about a point (x0, y0), the equations (2)will always define a one-to-one map of S0 on aregion I0 including the point (u0, v0). Thismap will represent approximately (, to in-finitesimals of higher order than the first) a pro-jection of the immediate neighborhood of (xQ, ya)on that of (n0, v0); in other words, strain—asmall circle about (.r0, y0) going approximatelyinto an ellipse about (u0, v0). If, however, u andv are conjugate functions, the ellipse becomes acircle, and the angle under which two curvesintersect in the (x, ;y)-plane is preserved in the(u, i>)-plane. Thus the shapes o
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Keywords: ., bookcentury1900, bookdecade1900, booksubjectencyclo, bookyear1903
