"Complex analysis... it's got so much magic in it."
"It's good, it's bad, it's great, it's awful, it's a lot of things, it's Vorinclex."
"The amazing thing is that conformal mappings preserve the solution of Laplace's equation."
"One of my favorite things to do is to be handed a complex question and work with a team of competent individuals to arrive at a solution."
"If a point eventually falls into some stable predictable pattern, we say that it's part of the Fatou set of our iterated function."
"It's just crazy to me they have more information than they're releasing and I wish I knew what it was."
"So that is the first half of the part one of the commander religious police gate multicolored commander review."
"We want to solve for where Z squared plus 1 is equal to 0 to find out what the poles are."
"If we know where the poles are, we can use Cauchy's residue theorem to muck around with some contours."
"As long as our contour encloses that pole, we can deform as much as we want, but the value will still stay the same."
"In fact, in the limit as R approaches infinity, this integral won't change."
"...every value of a complex function at every point carries information about every value of the complex function at every other point."
"...so the Bodie plot is really just a slice of the S plane, which I think is kind of cool how all of this stuff is interrelated."
"SU(2) is defined as the set of 2x2 complex unitary matrices with determinant 1."
"The concept of analytic continuation, Riemann zeta function, and Cauchy's famous residue theorem are beautiful ideas."
"Cauchy's residue theorem is perhaps the most fundamental result in complex analysis."
"If you're trying to take the integral along some curve in a region where the function is analytic, it doesn't matter what path you choose."
"The remarkable property of the zeta function is that it has an analytic continuation to the entire complex plane except for one point where s equals one."
"If f is holomorphic on U, and if f is not identically zero, then the zeros of f are discrete in U."
"Complex analysis is very much about holomorphic functions which are always analytic."
"The residue theorem, this is probably one of the most important formulas when you're studying complex analysis."
"Time translation in this notation is just a rotation in the complex plane."
"The famous Riemann hypothesis is the assertion that if the zeta function is 0 and the real part is between 0 & 1, you have to be on the line 1/2."
"Koshi's first integral theorem states that if f(z) is analytic in some simply connected region R or closed contour, then the integral over that path will be zero."
"Any nonzero polynomial has at least one root in C."
"The Riemann hypothesis is that all the non-trivial zeros of the Riemann zeta function have real part a half."
"Complex analysis is super important for partial differential equations and analysis."
"You can divide by zero in the complex plane."
"The Lao transform of a function is relatively simple to compute; it's related to the Fourier transform and it gives you a function in the complex plane."
"Complex analysis is also just good for you to know; it's like knowing calculus."
"The effect here is just to do a rotation counterclockwise by theta of the complex plane about the origin."
"This sequence converges to a unique limit not just in this circle over here whose radius is 1 over e, but it actually converges in an enormous region all the way down to minus e."
"The type of derivative that we are now going to introduce for complex functions is different in an amazing new way."
"The birth of complex analysis showed that complex numbers are important objects in solving real problems."
"Knowing something about analytic functions in the complex plane allows you to integrate a lot more stuff than you learned in Calculus class."
"If \( f \) is analytic and non-constant, then \( |f(z)| \) attains its maximum value on the boundary of the region."
"If z(t) equals x(t) plus iota y(t) is a continuous function in the interval [a, b], then geometrically it represents a curve in the xy-plane."
"A contour is a piecewise smooth arc."
"If f is analytic within and on a simple closed contour c, then the integral of f(z) along c is zero."
"The principle of deformation of path allows us to deform the path to a smaller contour."
"If f(z) is analytic at some point z0, then all derivatives of f(z) are also analytic at z0."
"If \( f'(z) \) is differentiable at every point inside, this means \( f'(z) \) is analytic at zero."
"If \( f(z) \) is analytic at that zero, then derivatives of all orders are also analytic at zero."
"If a function \( f \) is continuous in a domain \( D \), and integral of \( f(z) \) along every closed contour is zero, then \( f \) is analytic throughout \( D \)."
"Next up are some amazing consequences of Koshi theorem and integral formula; we'll see the maximum principle, Liouville's theorem, will be able to prove the fundamental theorem of algebra."
"A function that is a complex function can be split up into its real part U and its imaginary part V."
"The quest for understanding the zeta function contributed immensely to the development of complex analysis area of maths."
"The theory of analytic functions is closely tied to Laplace's equation."
"That's the crunch line of using complex numbers, e to the i theta is that they're absolutely great for taking powers."
"We discovered that the function \( f(z) \), defined by \( e^{x} \cos(y) + i e^{x} \sin(y) \), is an entire function."
"It's very hard to tackle using methods from real analysis, but it turns out by turning to complex analysis we can actually find this integral."
"A complex number naturally lives in something called the complex plane."
"Cauchy's theorem is the first indication that complex variables is a beautiful subject."
"That is how we can find out the complex form of Fourier series of any function \( f(x) \)."
"I just love seeing all the different ways of manipulating mathematical stuff in the complex plane."