MSS
001791 B
SCDIRB Theory of Functions.
I
References to Forsyth
by articles, thus : [ ]³
Cornell University, Summer 1904.
John Irwin Hutchinson
1) Interpretation (geometrical) of the complex
variable. z=x+iy=rεiθ Addition, subtraction, multiplication, intro-
preted geometrically by expressing in terms
of the coordinates (x,y), or (r,θ)) [1-3] Let w=f(z) and suppose that if z=x+iy w can be represented in the form w=u+iv. then δxδw=dzdw;δyδw=idzdw whence 2) 1) δxδu=δyδv;δyδu=−δxδv 3) [ This is the condition that any expression of the form]
u+iv can be expressed as of(z). These are also the conditions that dw shall be unique [that is that there shall be one
slope or tangent in the w-plane regardless
of the method of approach to a point on the slope
dxdy in the z plane]. For dzdw=dx+idydu+idv=1+iy′δxδu+δyδuy′+i(δxδv+δyδvy′){y′=dxdy To be independent of y, it is necessary for
the numerator to be divisible by 1+iy whence δxδu+iδxδv=(i1δyδu+δyδv) [Equating real [and imag^y parts]
and hence the above conditions.
(Picard II.n.2.)