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Messages - Jessica Long

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Chapter 2 / Section 2.2 "closed form" Qs
« on: October 12, 2020, 02:35:52 PM »
Questions 14-18 ask us to find a "closed form" for each power series. I assume this is a non-power series expression (e.g. ex). Some of the power series seem to be variants on the geometric series, but then the closed form would only hold for some z based on the value of |z|, depending on the series. Would it be ok to just specify that the solution only holds for some z?

Chapter 1 / Definition for limit is infinity?
« on: September 28, 2020, 07:39:28 PM »
I know we defined a limit at infinity, but what about when a limit at a point is infinity?
Would this definition work: For any 𝜀 > 0 there exists 𝛿 > 0 such that |z − z0| < 𝛿 ⇒ |f(z)| > 𝜀.
Or for limit at infinity is infinity: For any 𝜀 > 0 there exists 𝛿 > 0 such that |z| > 𝛿 ⇒ |f(z)| > 𝜀.

Chapter 1 / Section 1.3 Q15
« on: September 23, 2020, 10:57:04 PM »
I can see from the visual of the sets that D1 ∩ D2 is not connected, but I'm not sure how to prove this. I figure we would start by assuming the set is connected and there exists a polygonal curve in the set between some two points and then trying to derive a contradiction, but I don't know how to show the contradiction in this case.

Chapter 1 / Re: how to solve this problem?
« on: September 23, 2020, 06:03:07 PM »
This is my solution to the problem. Note that the result can also be derived from the Apollonius circle formula with 𝜌=0, since the y-axis bisects the line segment between (2,0) and (-2,0).

Chapter 1 / Re: Question on Textbook Section1.5 The Log Function Example 3
« on: September 22, 2020, 11:01:08 PM »
The derivative is with respect to n. I've attached my computation of the limit, it starts off in the same way as yours.

Edit: there should be an n2 in the denominator, but it does not affect the result.

Chapter 1 / Problems to 1.2 Q20
« on: September 22, 2020, 10:44:50 PM »
The question: Let z1 and z2 be distinct complex numbers. Show that the locus of points z={tz1+(1−t)z2,−∞<t<∞}, describes the line through z1 and z2. The values $01 give the line segment joining z1 and z2.

I have an intuitive understanding of why the locus is a line, as it is similar to the description of a line through two points in R^2. However, I'm not sure how to prove this holds in C. Should I be trying to express it as one of the equations for line in C, such as Re(n̄z) = C or w̄z + wz̄ = r?

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