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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.
ຕົວຢ່າງທີ່ໄດ້ເຮັດ · e^(i*pi)
Evaluate e^(i*pi)
e^{i \pi}
ຂັ້ນຕອນຕໍ່ຂັ້ນຕອນ
e^{i \pi} = -1
Power: E^(I·π) = -1.
ເປີດເຜີຍຄຳຕອບ
-1
ປັດຈຸບັນທ່ານ
ເລືອກບັນຫາ ຫຼື ພິມ ຫຼື ແຕ້ມເອງ. ທຸກບາດກ້າວ, ຮູບ, ຄໍາຕອບຖືກລົບຈົນກວ່າເຈົ້າຈະຖາມມັນ.
ຮັກສາການເຮັດວຽກຂອງທ່ານໄວ້
ບັນຊີຟຣີ ເພີ່ມຂໍ້ມູນໃສ່ບົດຮຽນທຸກບົດ, ບັນທຶກຂອງສິ່ງທີ່ທ່ານໄດ້ເຮັດແລ້ວ, ບັນຫາທີ່ທ່ານໄດ້ແກ້ໄຂໄວ້ໃນບ່ອນດຽວ ແລະ ຄູສອນທີ່ທ່ານສາມາດຖາມກ່ຽວກັບໜ້ານີ້. ຄະນິດສາດເອງແມ່ນເປີດໃຫ້ທຸກຄົນ, ບໍ່ວ່າຈະໄດ້ເຂົ້າສູ່ລະບົບ ຫຼື ບໍ່.
ລົງທະບຽນ
ចូល
ຕົວສະແດງທີ່ໃຊ້ຢູ່ທີ່ນີ້
ກົດຕົວອັກສອນໃດໜຶ່ງເພື່ອເບິ່ງຄວາມໝາຍເຕັມ, ຮູບ ແລະ ຕົວອັກສອນແຕ່ລະຕົວໃນມັນໝາຍຄວາມວ່າແນວໃດ.
\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.
ຕົວສະແດງໃນໜ້ານີ້
ກົດເພື່ອເບິ່ງຄວາມໝາຍຂອງມັນ
\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 ›
ວິທີການ: Functions of a complex variable
Power: E^(I·π) = -1.
ຄໍາຖາມທີ່ຄົນຖາມ
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.
ເພີ່ມເຕີມໃນ 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