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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · e^(i*pi)

Arithmetic · Arithmetic

Evaluate e^(i*pi)

e^{i \pi}

Àwọn ìṣàmúlò-ètò

e^{i \pi} = -1
Power: E^(I·π) = -1.

Fi àwọn àgbèwọlé hàn

-1

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Bii o ṣe le: Functions of a complex variable

Power: E^(I·π) = -1.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Real Analysis

Sequences and their limits Convergence of series Improper integrals Taylor approximation and error Continuity and differentiability, rigorously Sequences of functions and uniform convergence