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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.

Eżempju maħdum · e^(i*pi)

Arithmetic · Arithmetic

Evaluate e^(i*pi)

e^{i \pi}

Pass b'pass

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

    Power: E^(I·π) = -1.

Jiżvelaw it-tweġiba
-1

Issa inti Agħżel problema, jew ittajpja jew iġbed tiegħek stess.Kull pass, stampa, it-tweġiba moħbija sakemm inti tistaqsi.

Żomm tiegħek stess li jaħdmu

Kont b'xejn iżid noti fuq kull lezzjoni, rekord ta' dak li lest, problemi solvuti tiegħek f'post wieħed, u għalliem inti tista' tistaqsi dwar din il-paġna.Il-matematika nnifisha hija miftuħa għal kulħadd, log fil jew le.

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Taptap kwalunkwe simbolu għad-definizzjoni sħiħa, stampa, u dak li kull ittra fiha tfisser.

Kif għandek tuża: Functions of a complex variable

  1. Power: E^(I·π) = -1.

Mistoqsijiet li n-nies jistaqsu

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.

Aktar fil Real Analysis

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