Category:Analysis

B

• [×] Binomial Coefficients‎ (0)

C

• [+] Calculus‎ (5)
• [+] Complex Analysis‎ (8)
• [+] Continuity‎ (3)
• [×] Convergence‎ (0)

E

• [×] Exponential Function‎ (0)

F

• [×] Floor and Ceiling‎ (0)
• [×] Formulas for Pi‎ (0)
• [×] Fourier Analysis‎ (0)
• [×] Fractals‎ (0)
• [×] Function Theory‎ (0)
• [+] Functional Analysis‎ (3)

G

• [×] Gamma Function‎ (0)

G cont.

• [×] Generating Functions‎ (0)

H

• [×] Harmonic Analysis‎ (0)
• [×] Hyperbolic Functions‎ (0)

L

• [+] Limits‎ (3)
• [×] Limits of a Sequence of Sets‎ (0)
• [×] Limits of Functions‎ (0)
• [×] Limits of Sequences‎ (0)
• [×] Logarithms‎ (0)

M

• [+] Measure Theory‎ (1)

N

• [×] Number Bases‎ (0)
• [+] Numbers‎ (9)

P

• [+] Polynomial Theory‎ (1)
• [×] Power Series‎ (0)

R

• [+] Real Numbers‎ (1)

S

• [×] Sequences‎ (0)
• [+] Series‎ (4)
• [+] Special Functions‎ (1)

T

• [×] Taylor Series‎ (0)
• [+] Topology‎ (44)
• [×] Trigonometric Identities‎ (0)
• [×] Trigonometrical Series‎ (0)

U

• [×] Umbral Calculus‎ (0)

V

• [+] Vector Analysis‎ (1)

Z

• [×] Zeta Function‎ (0)

A

• Absolute Value Bounded Below by Zero
• Absolute Value is Functional
• Absolutely Continuous is Continuous
• Archimedean Principle

B

• Backwards Form of Triangle Inequality
• Banach Fixed-Point Theorem
• Basic Properties of Cosine Function
• Basic Properties of Sine Function
• Between Every Two Rationals Exists an Irrational
• Between Every Two Reals Exists a Rational
• Binet Form
• Bounded Set of Real Numbers
• Boundedness of Nth Powers
• Boundedness of Sine and Cosine
• Boundedness of Sine X over X
• Brouwer's Fixed Point Theorem/One-Dimensional Version

C

• Carathéodory's Theorem
• Cauchy's Convergence Criterion
• Cauchy's Inequality
• Cauchy-Schwarz Inequality
• Cauchy-Schwarz Inequality/Definite Integrals
• Change of Base of Logarithm
• Closure of Real Interval
• Concavity of Differentiable Functions
• Condition Defining Infimum
• Condition Defining Supremum
• Condition for Continuity on an Interval
• Constant Function is Uniformly Continuous
• Continuity of Composite Mapping
• Continuity of Root Function
• Continuity Property
• Continuous Function on Closed Interval is Uniformly Continuous
• Continuous Nowhere Differentiable Function
• Continuum Property
• Convergent Subsequence in Closed Interval
• Convex or Concave Function is Continuous
• Cosine Function is Absolutely Convergent
• Cosine Function is Continuous
• Cosine Function is Even
• Cosine of Multiple of Pi
• Cosine of Multiple of Pi Plus Half
• Cosine of Zero is One
• Cotangent Function is Periodic on Reals
• Countable Sets Have Measure Zero

D

• D'Alembert's Theorem
• Dedekind's Theorem
• Derivative of Geometric Progression
• Differentiable Bounded Convex or Concave Function is Constant
• Differentiable Function is Continuous
• Dirichlet's Test for Uniform Convergence
• Distance from Subset of Real Numbers
• Distance is a Metric

E

• Equivalence of Convex and Concave Definitions
• Equivalence of Definitions of Euler's Number
• Equivalence of Definitions of the Derivative
• Euler Formula for Sine Function
• Euler's Number is Irrational
• Euler's Number: Limit of Sequence implies Limit of Series
• Euler-Binet Formula
• Existence of Euler-Mascheroni Constant
• Existence of Logarithm

E cont.

• Existence of Root
• Exists Integer Below Any Real Number
• Exponent Combination Laws
• Exponent Combination Laws/Difference of Powers
• Exponent Combination Laws/Negative Power
• Exponent Combination Laws/Negative Power of Quotient
• Exponent Combination Laws/Power of Power
• Exponent Combination Laws/Power of Product
• Exponent Combination Laws/Power of Quotient
• Exponent Combination Laws/Sum of Powers

F

• Field of Real Numbers
• Fundamental Theorem of Algebra

G

• Gelfond-Schneider Theorem
• General Binomial Theorem

H

• Heine-Borel Theorem

I

• Image of Closed Real Interval is Bounded
• Image of Interval by Continuous Function
• Image of Interval by Derivative
• Inequalities Concerning Roots
• Infimum Plus Constant
• Infinite Limit Theorem
• Integral Power Function Bijective iff Index Odd
• Intermediate Value Theorem
• Intermediate Value Theorem for Derivatives
• Interval Defined by Absolute Value
• Interval Defined by Betweenness
• Interval Divided into Subsets
• Inverse of Convex Strictly Monotone Function
• Inverse of Positive Real is Positive
• Inverse of Strictly Monotone Function
• Irrationals Dense in Reals
• Iterative Process for Estimating Square Roots

K

• Kronecker's Lemma

L

• Lagrange Polynomial Approximation
• Laws of Logarithms
• Lebesgue Integral is Extension of Riemann Integral
• Limit iff Limits from Left and Right
• Limit of (Cosine (X) - 1) over X
• Limit of Function by Convergent Sequences
• Limit of Image of Sequence
• Limit of Sine of X over X
• Limit of Sine of X over X/Geometric Proof
• Limits of Convex or Concave Function
• Limsup Squeeze Theorem
• Linear Function is Continuous
• Linear Operator on General Logarithm
• Logarithms of Powers

M

• Max and Min of Function on Closed Real Interval
• Mean Value of Convex and Concave Functions
• Measurable Sets are a Sigma Algebra of Sets
• Minkowski's Inequality
• Modulus of Product
• Monotonic Additive Function is Linear
• Multiple of Infimum
• Multiple of Supremum

N

• Negative of Absolute Value
• Negative of Infimum
• Negative of Supremum

O

• Only Intervals are Connected
• Open Sets in Reals
• Order of the Moebius Function

O cont.

• Order Preserved on Positive Reals by Squaring
• Ordering of Squares in Reals
• Oscillation Zero if and only if Continuous

P

• Pi is Irrational
• Pointwise Addition forms a Group
• Polynomial is Continuous
• Powers Drown Logarithms

Q

• Quotient of Homogeneous Functions

R

• Rational Function is Continuous
• Rational Numbers are Close Packed
• Rational Numbers form Metric Space
• Rationals Dense in Reals
• Rationals form a Subfield of Reals
• Real Number Line is Complete Metric Space
• Real Number Line is Metric Space
• Real Numbers Between Epsilons
• Real Plus Epsilon
• Reals are Close Packed
• Restriction of Continuous Mapping
• Definition:Riemann Sum
• Root of Number Greater than One

S

• Second Derivative of Concave Downward Function
• Second Derivative of Concave Upward Function
• Set of All Real Intervals is Semiring of Sets
• Shape of Cosine Function
• Shape of Cotangent Function
• Shape of Sine Function
• Shape of Tangent Function
• Sine and Cosine are Periodic on Reals
• Sine and Cosine of Complementary Angles
• Sine and Cosine of Conjugate Angles
• Sine and Cosine of Supplementary Angles
• Sine Function is Absolutely Convergent
• Sine Function is Continuous
• Sine Function is Odd
• Sine of Multiple of Pi
• Sine of Multiple of Pi Plus and Half
• Sine of Multiple of Pi Plus Half
• Sine of Zero is Zero
• Square Number Less than One
• Stirling's Formula
• Strictly Monotone Function is Bijective
• Subspace of Real Continuous Functions
• Subspace of Real Functions of a Differentiability Class
• Subspace of Riemann Integrable Functions
• Sum of Arcsecant and Arccosecant
• Sum of Arcsine and Arccosine
• Sum of Logarithms
• Sum of Reciprocals is Divergent
• Suprema and Infima of Combined Bounded Functions
• Supremum Plus Constant

T

• Tangent Function is Odd
• Tangent Function is Periodic on Reals
• Tangent Line to Concave Up Graph
• Triangle Inequality

U

• Uniformly Continuous Function is Continuous
• Upper Bound of Natural Logarithm
• Upper Sum Never Smaller than Lower Sum

W

• Weierstrass M-Test

Z

• Zeroes of Sine and Cosine