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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.
Ohatra · e^(i*pi)
Dingana amin'ny dingana
- e^{i \pi} = -1
Power: E^(I·π) = -1.
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Ahoana ny fomba: Functions of a complex variable
- Power: E^(I·π) = -1.
Fanontaniana napetrak'ireo olona
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.
Mbola maro ao Real Analysis
Sequences and their limitsConvergence of seriesImproper integralsTaylor approximation and errorContinuity and differentiability, rigorouslySequences of functions and uniform convergence