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Unformatted text preview: 26 FUNCTIONS OF A COMPLEX l.l'ARIAIElLE We shall not dwell further on the ﬁner points of the mapping w = 22. Instead, we note
that in general it is extremely difﬁcult to plot images of curves in the 2 plane, even
the straight lines parallel to the axes, under polynomial mappings. We also note that we
do not often need to do so, and that we have done it only as an aid to understanding. The exercises that follow should also help in understanding this topic. We shall then
return to examine polynomial, rational and exponential mappings in Section 1.3.4, after
introducing complex differentiation. ' 1 ' 'Ii‘ ”.9... .ru:  u. . . I. . 12"". I,i _ I. _._. . :_ .___‘,  _: .‘i .r' . . LE”: . 1: ..r "4. i'l :_I I." 'I " . . I I .1; r, ' . . are at tee: . .~. 3."  ' t '. " wsf .ii . s    “ilkit
. . . " . I_. is “: _I Find the image region in the w plane corresponding
to the region inside the triangle in the a plane having vertices at O + jO, 2 + it} and 0 + j2 under the
mapping w = :2. Illustrate with sketches. iﬁi Find the images of the lines y = x and y = —x under h the mapping w = 2:2. Also ﬁnd the irnage of the
general line through the origin y = not. By putting
as = tan 6,, deduce that straight lines intersecting at
the origin in the a plane map onto'lines intersecting
at the origin in the w plane, but that the angle
between these image lines is double that between
the original lines. Consider the mapping w = z”, where a is an integer
(a generalization of the mapping w = zg). Use the (b) Straight lines passing through the origin
intersecting with angle 3,, in the a plane are
mapped onto straight lines passing through the
origin in the w plane but intersecting at an
angle nﬂﬂ. If the complex function 1+:2
2." W: is represented by a mapping from the a plane onto
the w plane, ﬁnd a! in terms ofx and y, and ti in terms
ofx and y, where a = x +jy, w = a +je. Find the
image of the unit circle s = 1 in the w plane. Show
that the circle centred at the origin, of radius r, in
the a plane (Isl: r) is mapped onto the curve polar representation of complex numbers to show 2 2 2 2
that (,W)+(j= (rel) r + 1 r — l
(a) Circles centred at the origin in the a plane are
mapped onto circles centred at the origin in the in the w plane. What kind of curves are these? What
w plane. happens for veryr large r?  Complex differentiation The derivative of a real function f (x) of a single real variable x at x = x, is given by the
limit foﬁ) = lim fitI) “.ﬂxo) xe—axﬂ x _ xﬂ Here, of course, m, is a real number and so can be represented by a single point on the
real line. The point representing .1: can then approach the ﬁxed so, either from the left or
from the right along this line. Let us now turn to complex variables and functions
depending on them. We know that a plane is required to represent complex numbers, so 2,, is now a ﬁxed point in the Argand diagram, somewhere in the plane. The deﬁnition
of the derivative of the function f (a) of the complex variable .3 at the point 2,, will thus be 34 FUNCTIONS OF A COMPLEX VARIABLE = 0
8x2 ayz and u is also a harmonic function. We have assumed that both at and 12 have continuous secondorder partial derivatives, so that
821)
8y8x (.921: = c921: 8212 aa aa’ an: Determine whether the following functions are
analytic, and ﬁnd the derivative where appropriate: (a) ze’ (b) sin4z
(c) zz* ((1) cos 22 Determine the constants a and b in order that w=x2+ayg—2xy+j(bx2—y2+2xy) be analytic. For these values of a and I) ﬁnd the
derivative of w, and express both w and dw/dz as
functions of z = x + j y. Find a function v(x, y) such that, given at = 2x(1 — y),
f(2) = u +jv is analytic in z. Sh0w that ¢(x, y) = e"(x cosy —y sin y) is a harmonic
function, and ﬁnd the conjugate harmonic function
W05, y). Write (Mac, y) + j y!(x, y) as a function of
z = x + j y only. Show that u(x, y) = sinx cosh y is harmonic. Find
the harmonic conjugate u(x, y) and express w = u +jv
as a function ofz = x +jy. Find the orthogonal trajectories of the following
families of curves: (at) xsy —xy3 = a (constant a) (b) e" cosy + xy = a (constant 05) Find the real and imaginary parts of the functions
(a) 2262‘
(b) SihZz Verify that they are analytic and ﬁnd their
derivatives. Give a deﬁnition of the inverse sine function
sin'lz for complex 2. Find the real and imaginary
parts of sin—‘2. (Hint: put 2 = sin w, split into real and imaginary parts, and with w = u + jy
andz=x+jy solve for u and v in terms ofx and y.) Is sin'1 2 analytic? If so, what is its
derivative? Establish that ifz=x+jy, sinhy[ s sinz $
coshy. 1.3.4 Mappings revisited In Section 1.2 we examined mappings from the 2 plane to the w plane, where in the
main the relationship between w and z, w = f(z) was linear or bilinear. There is an
important property of mappings, hinted at in Example 1.8 when considering the
mapping w = 22. A mapping w =f(z) that preserves angles is called conformal. Under
such a mapping, the angle between two intersecting curves in the 2 plane is the same as
the angle between the corresponding intersecting curves in the w plane. The sense of
the angle is also preserved. That is, if 8 is the angle between curves 1 and 2 taken in the
anticlockwise sense in the 2 plane then 9 is also the angle between the image of curve
1 and the image of curve 2 in the w plane, and it too is taken in the anticlockwise sense. 38 FUNCTIONS OF A COMPLEX VARIABLE —~————————____________ Figure 1.18 Mapping
of lines under w = e2. y = Im (z) 2 plane w plane (b) Since at2 + v2 = ez“, if x = 0 then a:2 + v2 = 1, so the imaginary axis in the 2 plane
corresponds to the unit circle in the w plane. If x < 0 then e” < l, and as x —> —oo, 6’" —> 0, so the left half of the 2 plane corresponds to the interior of the unit circle
in the w plane, as illustrated in Figure 1.19. Figure 1.19 Mapping
of halfplane under
w = e2. f .\ 
ﬁg: Determine the points at which the following
mappings are not conformal: (a) w=22—1 (b)w=223—2122+722+6 (c) w=8z+ i 222 Follow Example 1.13 for the mapping w = z — 1/z.
Again determine the points at which the mapping is
not conformal, but this time demonstrate this by
looking at the image of the imaginary axis. Find the region of the w plane corresponding to
the following regions of the 2 plane under the
exponential mapping w = e2: (a)0$x<m (b)0$x$l,0$y$l (c)%1t~<—ySTE,O<.x<oc Consider the mapping w = sin 2. Determine the
points at which the mapping is not conformal. By ﬁnding the images in the w plane of the lines x = constant and y = constant in the 2 plane
(2 2 x + jy), draw the mapping along similar lines to
Figures 1.14 and 1.18. Show that the transformation
a2
2 = + —
4 c where z = x + jy and I: = Re” maps a circle, with
centre at the origin and radius a, in the gplane, onto
a straight line segment in the 2 plane. What is the
length of the line? What happens if the circle in the
4: plane is centred at the origin but is of radius 13,
where b i a? ——————————.______,__________ ...
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 Fall '10
 ChinC.Lee

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