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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.

Kushanda muenzaniso · e^(i*pi)

Arithmetic · Arithmetic

Evaluate e^(i*pi)

e^{i \pi}

Step by step

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

    Power: E^(I·π) = -1.

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Maitiro: Functions of a complex variable

  1. Power: E^(I·π) = -1.

Mabvunzo vanhu vanobvunza

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.

More in Real Analysis

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