MATH 375 Mathematics Complex Equations

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Math 375 : Assignment 2

Due Thursday, Feb 21

Problem 1: Consider the complex function

f(x, y) = 

x

2

y

x

4 + y

2



+ i



xy2

x

2 + y

2



(a) Compute the limit of f as (x, y) ? (0, 0) along the vertical, imaginary axis:

lim

t ? 0

f(0, t)

(b) Compute the limit as (x, y) ? (0, 0) along any (every) other line, y = mx:

lim

t ? 0

f(t, mt)

(c) Your work above should establish the fact that the limit of f along any

straight line to the origin is the same. Despite this, prove that the (full,

two-dimensional, complex) limit does not exist. Hint: Try taking the limit

along a familiar curved path to the origin.

Problem 2: Consider the linear function: f(z) = (3 + 2i)z + (?4 + i)

(a) Prove, analytically, directly using the definition of limit, that

lim z ? 2?i

f(z) = 4 + 2i

Hint: We talked about a simple way to do this for linear functions. But the

more general ‘re-centering’ idea will also work here; see Problem 3. Try it

both ways!

(b) More concretely, if I give you an error radius  = 1 around 4 + 2i ? C, what

tolerance radius ? can you take around 2 ? i ? C that ensures

0 < |z ? (2 ? i)| < ? =? |f(z) ? (4 + 2i)| < 1

Explicitly test your answer for ? by picking a concrete point w in the

corresponding punctured tolerance disk around 2 ? i and confirming that

its image f(w) lands inside the error disk around 4 + 2i of radius  = 1.

(Compute f(w) explicitly using the formula!) Draw a picture of the

punctured tolerance disk around 2 ? i, showing your point w, and draw

a second picture of the error disk, showing the image f(w). Finally, also

explain your answer for ? geometrically, using the fact that f is linear.

Math 375 : Assignment 2 Due Thursday, Feb 21 Problem 1: Consider the complex function  2    xy xy 2 f (x, y) = +i 2 x4 + y 2 x + y2 (a) Compute the limit of f as (x, y) → (0, 0) along the vertical, imaginary axis: lim f (0, t) t→0 (b) Compute the limit as (x, y) → (0, 0) along any (every) other line, y = mx: lim f (t, mt) t→0 (c) Your work above should establish the fact that the limit of f along any straight line to the origin is the same. Despite this, prove that the (full, two-dimensional, complex) limit does not exist. Hint: Try taking the limit along a familiar curved path to the origin. Problem 2: Consider the linear function: f (z) = (3 + 2i)z + (−4 + i) (a) Prove, analytically, directly using the definition of limit, that lim f (z) = 4 + 2i z → 2−i Hint: We talked about a simple way to do this for linear functions. But the more general ‘re-centering’ idea will also work here; see Problem 3. Try it both ways! (b) More concretely, if I give you an error radius  = 1 around 4 + 2i ∈ C, what tolerance radius δ can you take around 2 − i ∈ C that ensures 0 < |z − (2 − i)| < δ =⇒ |f (z) − (4 + 2i)| < 1 Explicitly test your answer for δ by picking a concrete point w in the corresponding punctured tolerance disk around 2 − i and confirming that its image f (w) lands inside the error disk around 4 + 2i of radius  = 1. (Compute f (w) explicitly using the formula!) Draw a picture of the punctured tolerance disk around 2 − i, showing your point w, and draw a second picture of the error disk, showing the image f (w). Finally, also explain your answer for δ geometrically, using the fact that f is linear. Problem 3: Consider the function f (z) = z 2 (a) Prove, directly from the definition, that lim f (z) = 2i z → 1+i Hint: Start by ‘re-centering’ the formula at z0 = 1+i. Recall that the general idea is as follows: For any z0 ∈ C, we of course have z = (z − z0 ) + z0 . In your formula for f (z), you can then replace each z by [(z − z0 ) + z0 ], then expand, always keeping (z − z0 ) as a single unit. For a quadratic function, this allows you to write f in the form f (z) = a(z − z0 )2 + b(z − z0 ) + c, for some complex constants a, b, c ∈ C. In this case, you have f (z) = z 2 , and you want to replace each z in the formula, (there is only one), with z = [z − (1 + i)] + (1 + i) Square this, but always keep [z − (1 + i)] as a single unbroken unit. After you do that, move on to the proof. Start by showing all the ‘preliminary work’ where you work ‘backwards’ to determine δ (in terms of ). Finally, once you have δ, write the proof, from beginning to end, going ‘forwards’. Starting with the line “Fix any  > 0.” (We did this in class! Write it out like we did in class.) (b) More concretely, if I give you an error radius  = 1/100 around L = 2i, what tolerance radius δ can you take around z0 = 1 + i that ensures 0 < |z − (1 + i)| < δ =⇒ |z 2 − 2i| < 1/100 (c) Now consider a more general quadratic function f (z) = (1 − i)z 2 + (−2 + i)z + (7 − i) Prove, analytically, using the definition, that lim f (z) = −5 + 2i z → 3i Hint: Use the re-centering idea! Start by rewriting each z = [(z − 3i) + 3i], then expand, always keeping (z − 3i) as a single unit. Show all your work and write the proof as explained in part (a).

Tutor Answer

Super_Teach12
School: Cornell University

Hi, here is your assignment :). Everything has done through step by step solution :)

Math 375 : Assignment 2
Problem 1:
a. Along the vertical, imaginary axis:
Given the function, if is along the x-axis then;
lim

(𝑥,𝑦)→(𝑥,0)

𝑥2𝑦
( 4
)=𝟎
(𝑥,𝑦)→(𝑥,0) 𝑥 + 𝑦 2...

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Anonymous
Tutor went the extra mile to help me with this essay. Citations were a bit shaky but I appreciated how well he handled APA styles and how ok he was to change them even though I didnt specify. Got a B+ which is believable and acceptable.

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