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Functions of a complex variable

Euler's formula, and why complex analysis is where the Riemann Hypothesis lives.

e^{iθ} = cos θ + i sin θ turns rotation into exponentiation, and it is the first step into a subject where differentiable functions are astonishingly rigid: knowing one on a small disc determines it everywhere. The zeta function ζ(s) of the Riemann Hypothesis is such a function; its zeros are the whole story.

Mfano wenye matokeo: e^(i*pi)

Arithmetic · Arithmetic

Evaluate e^(i*pi)

-1

Hatua kwa hatua

  1. e^{i \pi} = -1

    Power: E^(I·π) = -1.

Lafunua jibu
-1

Symbols used here

\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
i
imaginary unit
i² = −1.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
\infty
infinity
Not a number: "grows without bound" in limits and intervals.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\lim_{x \to a} f(x)
limit
The value f(x) approaches as x approaches a.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
\forall,\ \exists
for all, there exists
Quantifiers: every x; at least one x.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
\varepsilon,\ \delta
epsilon, delta
Small positive tolerances in the definition of a limit.
\sup,\ \inf
supremum, infimum
Least upper bound, greatest lower bound.
B(x, r),\ \overline{A},\ \partial A
open ball, closure, boundary
Points within r of x; A plus its limit points; the edge of A.

How to: Functions of a complex variable

  1. Power: E^(I·π) = -1.

Questions people ask

What is the ε–δ definition actually saying?

That you can make the output as close to the limit as anyone demands (within ε) by keeping the input close enough (within δ). It replaces "approaches" with a challenge-and-response that can be checked.

Why does the harmonic series diverge when its terms go to zero?

Because the terms shrink too slowly: group them as 1/3 + 1/4 > 1/2, 1/5 + … + 1/8 > 1/2, and so on — infinitely many halves.

Jaribu kufanya mambo yako mwenyewe

Mengi zaidi katika Real Analysis

Sequences and their limitsConvergence of seriesImproper integralsTaylor approximation and errorContinuity and differentiability, rigorouslySequences of functions and uniform convergence