What is Complex analysis: Definition and 778 Discussions
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.As a differentiable function of a complex variable is equal to its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions).
So I really have a few questions. First, is it wise to take the following classes in the same semester?
Quantum Mechanics I - At the level of Griffiths' Intro to Quantum Mechanics, chapters 1-5ish
Classical Dynamics - At the level of Thorton and Marion, Chapters 1-12, ends with coupled...
can you recommend a good book on complex analysis? I would like a book that can sharpen my skills in solving complex number problems through graphs and also improve the algebraic part like solving problems related to roots of unity etc.
(I have studied calculus myself. I have done a lot of self...
what is the nature of singularity of the function f(x)=exp(-1/z) where z is a complex number?
now i arrive at two different results by progressing in two different ways.
1) if we expand the series f(z)=1-1/z+1/2!(z^2)-... then i can say that z=0 is an essential singularity.
2) now again if i...
Homework Statement
I have uploaded necessary image(s) for the question I have successfully accomplished a, but I am not sure how to start b.
Homework Equations
The sum of the integral paths added up = the desired result.
The Attempt at a Solution
[/B]
So we start with path CR
And then go...
Homework Statement
Computer the integral:
Integral from 0 to infinity of (d(theta)/(5+4sin(theta))
Homework Equations
integral 0 to 2pi (d(theta)/1+asin(theta)) = 2pi/(sqrt(1-a^2)) (-1<a<1)
The Attempt at a Solution
I've seen this integral be computed from 0 to 2pi, where the answer is 2pi/3...
Homework Statement
Problem and solution found here: http://homepages.math.uic.edu/~dcabrera/math417/summer2008/section57_59.pdf
The question I am interested in is #1. In the solution, the instructor differentiates the series to get to:
2/(1-z)^3 = the series.
If I want the Maclaurin series of...
Homework Statement
Let 0 < r < 1. Show that
from n=1 to n=∞ of Σ(r^ncos(n*theta)) = (rcos(theta)-r^2)/(1-2rcos(theta)+r^2)
Hint. This is an example of the statement that sometimes the fastest path to a “real” fact is via complex numbers. Let z = reiθ. Then, since r = |z|, and 0 < r < 1, the...
Homework Statement
Express (-1)1/10 in exponential form
(My first time posting - I hope I got the syntax right!)
Homework Equations
The Attempt at a Solution
[/B]
I got the solution, it's ejπ/10, but I'm not sure why. Here's my work:
(-1)1/10 = (cos(π) + jsin(π))1/10 = cos(pi/10) +...
Homework Statement
With . Give an example, if it exists, of a non constant holomorphic function that is zero everywhere and has the form 1/n, where n € N.
Homework Equations
So.. This was in my Complex Analysis exam, and i have no idea what to do. I always seem to get stuck at these more...
Hi all,
I was unsure where to put this thread as I read the main topic title in the topology/analysis forums and decided to post it here.
I am looking for a chart/graph/website that helps me understand the basic terms such as:
-neighborhoods
-Boundary points
-Singularity points
- "Function is...
I am currently learning how to work with Cauchy-Riemann equations.
The equation is f(z) = 2x+ixy^2.
My question: is u(x,y) = 2x or just x?
At this link: http://www.math.mun.ca/~mkondra/coan/as3a.pdf in letter e) they say u(x,y) is equal to x. But I don't understand how that is possible.
Is...
Homework Statement
Find the image of the rectangle with four vertices A=0, B= pi*i, C= -1+pi*i, D = -1 under the function f(z)=e^x
2. The attempt at a solution
So, the graph of the original points is obvious.
Now I have to map them to the new function.
Seems easy enough, but I am not getting...
The problem states, Show that:
a) |e^(i*theta)| = 1.
Now, the definition of e^(i*theta) makes this
|cos(theta)+isin(theta)|
If we choose any theta then this should be equal to 1.
What can help me prove this? If I choose, say, pi/6 then it simplifies to |(sqrt(3))/2+i/2)| which doesn't seem to...
Use properties to show that:
(question is in the attached picture)
Now, it is my understanding that due to properties you can express (sqrt(5)-i) as the sqrt((sqrt(5))^2+(-1)^2) which equals sqrt(6).
And (2zbar+5) can be represented as (2z+5).
But this would be sqrt(6)*(2z+5) which is NOT...
Sorry if this is the wrong forum to post this-
Can anyone suggest a good (ideally online) resource for challenging complex analysis problems? The ones I have found so far have been mainly computational- I'm looking for conceptually harder problems, preferably requiring lots of proofs, which...
Homework Statement
Evaluate the integral using any method:
∫C (z10) / (z - (1/2))(z10 + 2), where C : |z| = 1
Homework Equations
∫C f(z) dz = 2πi*(Σki=1 Resp_i f(z)
The Attempt at a Solution
Rewrote the function as (1/(z-(1/2)))*(1/(1+(2/z^10))). Not sure if Laurent series expansion is the...
Here's my situation:
Summer 2015, I am majoring in math and physics.
I am taking a 4-week course on DIFF EQ right now, and completely loving it and doing extremely well. Just finished my set of Calc 1, 2, and 3, and an intro to advanced math course (proof-writing basics).
Diff EQ is a 220...
Let G be a non-empty open connected set in Rn, f be a differentiable function from G into R, and A be a linear transformation from Rn to R. If f '(a)=A for all a in G, find f and prove your answer.
I thought of f as being the same as the linear transformation, i.e. f(x)=A(x). Is this true?
http://www.math.hawaii.edu/~williamdemeo/Analysis-href.pdf
Please look at problem 2 on page 39 of the problems/solutions linked above.
I know I'm going to kick myself when someone explains this to me but how was equation "(31)" of the solution obtained? The first term of the RHS of (31) is...
I need recommendation about complex analysis book. As I'm electrical eng. student, it should cover everything one engineer need to know about that mathematical field, but without strict mathematical formalism :)
Homework Statement
Show that if
P(z)=a_0+a_1z+\cdots+a_nz^n
is a polynomial of degree n where n\geq1 then there exists some positive number R such that
|P(z)|>\frac{|a_n||z|^n}{2}
for each value of z such that |z|>R
Homework Equations
Not sure.
The Attempt at a Solution
I've tried dividing...
Here's a link to a professor's notes on a contour integration example.
https://math.nyu.edu/faculty/childres/lec12.pdf
I don't understand where the ##e^{i\pi /2} I## comes from in the first problem. It seems like it should be ##e^{i\pi}## instead since ##-C_3## and ##C_1## are both on the real...
I'm going to ask a very general question where I just would want to hear different possible methods that can be thought of in this kind of problem. I am trying to solve a very specific problem with this but I won't talk about that because I don't want someone to give me the answer but ideas for...
Hello,
I am trying to understand how to get the residue as given by wolfram :
http://www.wolframalpha.com/input/?i=residue+of+e^{Sqrt[x^2+%2B+1]}%2F%28x^2+%2B+1%29^2
The issue I am facing is - since it is a second order pole, when I try to different e^{\sqrt{x^+1}} I get a \sqrt{x^+1}...
How do you integrate f(z) = e^(1/z) in the multiply connected domain {Rez>0}∖{2}
It seems like integrals of this function are path independent in this domain since integrals of e^(1/z) exist everywhere in teh domain {Rez>0}∖{2}. Is that correct?
Homework Statement
In each case, state whether the assertion is true or false, and justify your answer with a proof or counterexample.
(a) Let ##f## be holomorphic on an open connected set ##O\subseteq \mathcal{C}##. Let ##a\in O##. Let ##\{z_k\}## and ##\{\zeta_k\}## be two sequences...
Are the integrals of the function f(z) = (1/(z-2) + (1/(z+1) + e^(1/z)
path independent in the following domain: {Rez>0}∖{2}
The domain is not simply connected
I know that path independence has 3 equivalent forms
that are
1) Integrals are independent if for every 2 points and 2 contours...
Assume |f(z)| >= 1/3|e^(z^2)| for all z in C and that f(0) = 1 and that f(z) is entire. Prove that f(z) = e^(z^2) for all z in C.
How do you start for this.
Hi,
In my textbook the following theorem is designated "Proposition 3.4.2 part (vi)". There are 6 parts total in the overall theorem. I'll just type the part I'm interested in below. My question is, is there a more standard name for this theorem? I would like to find an additional...
Homework Statement
Find the roots of z^4+4=0 and use that to factor the expression into quadratic factors with real coefficients.
Homework Equations
DeMoivre's formula.
The Attempt at a Solution
I have been able to identify they are \pm 1 \pm i but i have no idea how to factor the...
On page 671 Mary Boas has her Theorem III for that chapter. Roughly it tells us that if f(z) -a complex function- is analytic in a region, inside that region f(z) has derivatives of all orders. We can also expand this function in a taylor series.
I get the part about a Taylor series, that's...
Homework Statement
For ##|z-a|<r## let ##f(z)=\sum_{n=0}^{\infty}a_n (z-a)^n##. Let ##g(z)=\sum_{n=0}^{\infty}b_n(z-a)^n##. Assume ##g(z)## is nonzero for ##|z-a|<r##. Then ##b_0## is not zero.
Define ##c_0=a_0/b_0## and, inductively for ##n>0##, define
$$
c_n=(a_n - \sum_{j=0}^{n-1} c_j...
Homework Statement
Let f be a function with a power series representation on a disk, say D(0,1). In each case, use the given information to identify the function. Is it unique?
(a) f(1/n)=4 for n=1,2,\dots
(b) f(i/n)=-\frac{1}{n^2} for n=1,2,\dots
A side question:
Is corollary 1 from my...
Homework Statement
Give an example of a power series with [itex]R=1[\itex] that converges uniformly for [itex]|z|\le 1[\itex], but such that its derived series converges nowhere for [itex]|z=1|[\itex].
Homework Equations
R is the radius of convergence and the derived series is the term by term...
Homework Statement
Find the largest set D on which f(z) is analytic and find its derivative. (If a branch is not specified, use the principal branch.)
f(z) = Log(iz+1) / (z^2+2z+5)
Homework EquationsThe Attempt at a Solution
Not sure how to even attempt this solutions but I wrote down that...
Homework Statement
This isn't a standard homework problem. We were asked to do research and to find a theorem of the form:
If something about the partial derivatives of u and v is true then the implication is ##D(u,v)## at ##(x_0,y_0)## exists from ##R^2## to ##R^2##Homework EquationsThe...
Homework Statement
Hi, I'm stuck with this question:
How many branches (solutions) and branch points does the function
f(z) = (z2 +1 +i)1=4 have? Give an example of a branch of the multi-
valued function f that is continuous in the cut-plane, for some choice
of branch cut(s). Now by choosing...
Homework Statement
Describe the set of points determined by the given condition in the complex plane:
|z - 1 + i| = 1
Homework Equations
|z| = sqrt(x2 + y2)
z = x + iy
The Attempt at a Solution
Tried to put absolute values on every thing by the Triangle inequality
|z| - |1| + |i| = |1|...
Homework Statement
Find the residue of:
$$f(z) = \frac{(\psi(-z) + \gamma)}{(z+1)(z+2)^3} \space \text{at} \space z=n$$
Where $n$ is every positive integer because those $n$ are the poles of $f(z)$Homework EquationsThe Attempt at a Solution
This is a simple pole, however:
$$\lim_{z \to n}...
Homework Statement
Find the residue at z=-2 for
$$g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$
Homework Equations
$$\psi(-z)$$ represents the digamma function, $$\zeta(z)$$ represents the Riemann-Zeta-Function.
The Attempt at a Solution
I know that:
$$\psi(z+1) = -\gamma - \sum_{k=1}^{\infty}...
Homework Statement
Homework Equations
Down
The Attempt at a Solution
As you see in the solution, I am confused as to why the sum of residues is required.
My question is the sum:
$$(4)\cdot\sum_{n=1}^{\infty} \frac{\coth(\pi n)}{n^3}$$
Question #1:
-Why is the beginning n=1 the residue...
How relevant is complex analysis to physics? I really want to take differential equations but I would have to change my schedule around way more than I want to. So, would anyone advise a physics major to to take complex analysis? Should I just change my schedule around so I can take differential...
I posted the same question on Math Stackexchange: http://math.stackexchange.com/questions/1084724/calculating-harmonic-sums-with-residues/1085248#1085248
The answer there using complex analysis is great. I had questions, which Id like to get advice on here.
(1) How did he get the laurent...
I saw this method of calculating:
$$I = \int_{0}^{1} \log^2(1-x)\log^2(x) dx$$
http://math.stackexchange.com/questions/959701/evaluate-int1-0-log21-x-log2x-dx
Can you take a look at M.N.C.E.'s method?
I don't understand a few things.
Somehow he makes the relation...
Hello,
I am evaluating:
$$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$
Using the following contour:
$R$ is the big radius, $\epsilon$ is small radius (of small circle)
Question before: Which $\log$ branch is this? I asked else they said,
$$-\pi/2 \le arg(z) \le 3\pi/2$$
But in the...
Consider the integral:
$$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$
$R$ is the big radius, $\delta$ is the small radius.
Actually, let's consider $u$ the small radius. Let $\delta = u$
Ultimately the goal is to let $u \to 0$
We can parametrize,
$$z =...
Homework Statement
First, let's take a look at the complex line integral.
What is the geometry of the complex line integral?
If we look at the real line integral GIF:
[2]: http://en.wikipedia.org/wiki/File:Line_integral_of_scalar_field.gif
The real line integral is a path, but then you...
I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a circle segment. The closest I can find is the Schwarz-Christoffel mapping.
Anyone has any tips?
Homework Statement
Show that the real and imaginary parts of the following susceptibility function satisfy the K-K relationships. Use the residue theorem.
$$ \chi(\omega) = \frac{\omega_{p}^2}{(\omega_0^2-\omega^2)+i\gamma\omega} $$
Homework Equations
The Kramers-Kronig relations are
$$...