2. (a) Suppose that f is a continuous real-valued function defined on a closed rectangle [a, b] × [c, d]. Prove that if f takes on the values f(z) 25. Complex Functions 517 and f(w) for z and w in [a, b] × [c, d], then f also takes all values between f(z) and f(w). Hint: Consider g(t) = f(1z + (1 – in [0, 1]. *(b) If f is a continuous complex-valued function defined on [a, b] × [6, d], the assertion in part (a) no longer makes any sense, since we cannot talk of complex numbers between f(z) and f(w). We might t)w) for t iecture the line betwe
2. (a) Suppose that f is a continuous real-valued function defined on a closed rectangle [a, b] × [c, d]. Prove that if f takes on the values f(z) 25. Complex Functions 517 and f(w) for z and w in [a, b] × [c, d], then f also takes all values between f(z) and f(w). Hint: Consider g(t) = f(1z + (1 – in [0, 1]. *(b) If f is a continuous complex-valued function defined on [a, b] × [6, d], the assertion in part (a) no longer makes any sense, since we cannot talk of complex numbers between f(z) and f(w). We might t)w) for t iecture the line betwe
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please solve question 2 completely
![LTE
7:29 PM 4
VOLTE leO
46%
PROBLEMS
1. (a) For any real number y, define a(x) = x + iy (so that a is a complex-
valued function defined on R). Show that a is continuous. (This
follows immediately from a theorem in this chapter.) Show similarly
that B(y) = x + iy is continuous.
(b) Let f be a continuous function defined on C. For fixed y, let g(x) =
f(x + iy). Show that g is a continuous function (defined on R). Show
similarly that h(y) = f(x + iy) is continuous. Hint: Use part (a).
(a) Suppose that f is a continuous real-valued function defined on a
closed rectangle [a, b] × [c, d]. Prove that if f takes on the values f(z)
2.
25. Complex Functions 517
and f(w) for z and w in [a, b]× [c, d], then f also takes all values
between f(2) and f(w). Hint: Consider g(t) = f(tz + (1 – t)w) for t
in [0, 1].
*(b) If f is a continuous complex-valued function defined on [a, b] ×
[c, d], the assertion in part (a) no longer makes any sense, since we
cannot talk of complex numbers between f(2) and f(w). We might
conjecture that f takes on all values on the line segment between f(z)
and f(w), but even this is false. Find an example which shows this.
3. (a) Prove that if ao, . .. , an-1 are any complex numbers, then there are
, žn (not necessarily distinct) such that
complex numbers z1,
z" + an-12"-1+ · · · + ao =
II (z - z:).
i=1
(b) Prove that if ao, . .., an-1 are real, then z" + an-12*=1 + • · · + ao
can be written as a product of linear factors z + a and quadratic
factors z? + az + b all of whose coefficients are real. (Use Problem
24-7.)
4. In this problem we will consider only polynomials with real coefficients.
Such a polynomial is called a sum of squares if it can be written as
hi? + ··· + hn² for polynomials h; with real coefficients.
(a) Prove that if f is a sum of squares, then f(x) > 0 for all x.
(b) Prove that if f and g are sums of squares, then so is f g.
(c) Suppose that f(x) > 0 for all x. Show that f is a sum of squares. Hint:
First write f(x) = x*g(x), where g(x) + 0 for all x. Then k must be
even (why?), and g(x) > 0 for all x. Now use Problem 3(b).
5. (a) Let A be a set of complex numbers. A number z is called, as in the real
case, a limit point of the set A if for every (real) & > 0, there is a
point a in A with |2 – al < e but z # a. Prove the two-dimensional
version of the Bolzano-Weierstrass Theorem: If A is an infinite subset
of {a, b] × [c, d], then A has a limit point in [a, b] × [c, d]. Hint:
First divide [a, b] × [c, d] in half by a vertical line as in Figure 7(a).
Since A is infinite, at least one half contains infinitely many points of
A. Divide this in half by a horizontal line, as in Figure 7(b). Continue
in this way, alternately dividing by vertical and horizontal lines.
(The two-dimensional bisection argument outlined in this hint is so](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4654d3a-a61f-43c1-b797-b818a0cf5d64%2F8ed8bf60-0dba-4175-98cc-93ef4f517a6a%2Fo9p44n2_processed.png&w=3840&q=75)
Transcribed Image Text:LTE
7:29 PM 4
VOLTE leO
46%
PROBLEMS
1. (a) For any real number y, define a(x) = x + iy (so that a is a complex-
valued function defined on R). Show that a is continuous. (This
follows immediately from a theorem in this chapter.) Show similarly
that B(y) = x + iy is continuous.
(b) Let f be a continuous function defined on C. For fixed y, let g(x) =
f(x + iy). Show that g is a continuous function (defined on R). Show
similarly that h(y) = f(x + iy) is continuous. Hint: Use part (a).
(a) Suppose that f is a continuous real-valued function defined on a
closed rectangle [a, b] × [c, d]. Prove that if f takes on the values f(z)
2.
25. Complex Functions 517
and f(w) for z and w in [a, b]× [c, d], then f also takes all values
between f(2) and f(w). Hint: Consider g(t) = f(tz + (1 – t)w) for t
in [0, 1].
*(b) If f is a continuous complex-valued function defined on [a, b] ×
[c, d], the assertion in part (a) no longer makes any sense, since we
cannot talk of complex numbers between f(2) and f(w). We might
conjecture that f takes on all values on the line segment between f(z)
and f(w), but even this is false. Find an example which shows this.
3. (a) Prove that if ao, . .. , an-1 are any complex numbers, then there are
, žn (not necessarily distinct) such that
complex numbers z1,
z" + an-12"-1+ · · · + ao =
II (z - z:).
i=1
(b) Prove that if ao, . .., an-1 are real, then z" + an-12*=1 + • · · + ao
can be written as a product of linear factors z + a and quadratic
factors z? + az + b all of whose coefficients are real. (Use Problem
24-7.)
4. In this problem we will consider only polynomials with real coefficients.
Such a polynomial is called a sum of squares if it can be written as
hi? + ··· + hn² for polynomials h; with real coefficients.
(a) Prove that if f is a sum of squares, then f(x) > 0 for all x.
(b) Prove that if f and g are sums of squares, then so is f g.
(c) Suppose that f(x) > 0 for all x. Show that f is a sum of squares. Hint:
First write f(x) = x*g(x), where g(x) + 0 for all x. Then k must be
even (why?), and g(x) > 0 for all x. Now use Problem 3(b).
5. (a) Let A be a set of complex numbers. A number z is called, as in the real
case, a limit point of the set A if for every (real) & > 0, there is a
point a in A with |2 – al < e but z # a. Prove the two-dimensional
version of the Bolzano-Weierstrass Theorem: If A is an infinite subset
of {a, b] × [c, d], then A has a limit point in [a, b] × [c, d]. Hint:
First divide [a, b] × [c, d] in half by a vertical line as in Figure 7(a).
Since A is infinite, at least one half contains infinitely many points of
A. Divide this in half by a horizontal line, as in Figure 7(b). Continue
in this way, alternately dividing by vertical and horizontal lines.
(The two-dimensional bisection argument outlined in this hint is so
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