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Taylor approximation and error
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Continuity and differentiability, rigorously
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.
Isibonelo esisebenza · e^(i*pi)
Evaluate e^(i*pi)
e^{i \pi}
Isigaba
e^{i \pi} = -1
Power: E^(I·π) = -1.
Bonisa impendulo
-1
Manje
Khetha izinkinga, noma ubhale noma udwebe zakho. Isigaba ngasinye, isithombe, impendulo ifihlakele kuze kube ubuza.
Qhubeka usebenza
I-akhawunti emahhala ifaka iziphawulo kulesifundo ngasinye, irekhodi lokuthi yini oqedile, izinkinga zakho ezixazululiwe endaweni eyodwa, nodokotela owaziyo ongabuza mayelana nalekhasi. I-matematics ivulekile kubo bonke, noma ngabe babhalise noma cha.
Bhala
Ngena
Amaphawu asetshenziswa lapha
Chofoza noma yiluphi uphawu lwesibonakaliso ukuze uthole ukuchaza okuphelele, isithombe, kanye nokuthi iyiphi inhlamvu ekhona ekhuluma ngaso.
\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 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) integral Antiderivative (indefinite) or signed area from a to b (definite). \mathbb{N} number sets Naturals, integers, rationals, reals, complex numbers. \forall 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 epsilon, delta Small positive tolerances in the definition of a limit. \sup supremum, infimum Least upper bound, greatest lower bound. B(x open ball, closure, boundary Points within r of x; A plus its limit points; the edge of A.
Amaphawu kulesikhasi
Chofoza enye ukuze ubone ukuthi iyini inkinga
\pi pi › e Euler's number › i imaginary unit › \leq less/greater than or equal › |x| absolute value / modulus › \infty infinity › \sum_{k=1}^{n} a_k summation › \prod_{k=1}^{n} a_k product › \lim_{x \to a} f(x) limit › \int f(x) integral › \mathbb{N} number sets › \forall for all › \blacksquare\ \text{or}\ \square end of proof › \varepsilon epsilon › \sup supremum › B(x open ball ›
Indlela: Functions of a complex variable
Power: E^(I·π) = -1.
Imibuzo abantu bebuza
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.
Okuningi 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