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).
Homework Statement
Homework Equations
The relevant equation is that sqrt(z) = e^(1/2 log z) and the principal branch is from (-pi, pi]
The Attempt at a Solution
The solution is provided, since this isn't a homework problem (I was told to post it here anyway). I don't understand why the...
(mentor note: this is a homework problem with a solution that the OP would like to understand better)
In Taylor's Complex Variables,
Example 1.4.10
Can someone help me understand this? I don't know what they mean by (i, i inf), or how they got it and -it
I'm learning complex analysis right now, and I'm reading from Joseph Taylor's Complex Variables.
On Theorem 1.4.8, it says "If a log is the branch of the log function determined by an interval I, then log agrees with the ordinary natural log function on the positive real numbers if and only if...
<Moderator's note: moved from a technical forum, so homework template missing>
Hi. I have solved the others but I am really struggling on 22c. I need it to converge for |z|>2. This is the part I am really struggling with. I am trying to get both fractions into a geometric series with...
I'm currently an applied math major. I'm creating a schedule for my next semester and I have the choice to take either complex variables or vector analysis with linear algebra and a college geometry course(elective of choice), but I don't know which pairing will be less stressful. I am currently...
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|...
Some calculators say (-2)2/3 is equal to ##-\frac{1}{2^\frac{1}{3}}+i\frac{3^\frac{1}{2}}{2^\frac{1}{3}}## while others say its equal to ##4^{\frac{1}{3}}## i.e. ##|-\frac{1}{2^\frac{1}{3}}+i\frac{3^\frac{1}{2}}{2^\frac{1}{3}}|##.
I think I am right to imply from above that (-2)2/3 does have an...
I have seen in the online Stanford Encyclopedia of Philosophy in the entry on Copenhagen Interpretation of Quantum Mechanics that Niels Bohr had argued that the theory of relativity is not a literal representation of the universe:
"Neither does the theory of relativity, Bohr argued, provide us...
Homework Statement
Find the Laurent series expansion of f(z) = \log\left(1+\frac{1}{z-1}\right) in powers of \left(z-1\right).
Homework Equations
The function has a singularity at z = 1, and the nearest other singularity is at z = 0 (where the Log function diverges). So in theory there should...