User:Keith.U
Undergraduate mathematics\statistics student. Areas of study: Real analysis, metric spaces, probability theory.
My Current Focus
My current focus on proofwiki is developing a well-rounded and rigorous exposition of the exponential and (necessarily) logarithmic functions. Some of the approaches are rather tedious, and thus I'm forced to make my own proofs for much of the minutia, a recipe for certain disaster. If I've made a mistake somewhere, or you think I could be approaching a certain subject with more ease, please let me know!
Pages I've Created or Refactored
This exists as a quick reference for me to check over previous pages I've created or done major edits on, so that I may keep track of what I've done and round out entries made while perhaps overtired. If a page you have created or done significant work on appears here, please don't take this to mean I'm trying to claim any credit for your work. Again, this is so I may easily check for any "proofread", "tidy" or "refactor" tags that might arise, so I may correct my mistakes. A star indicates that it is up to house standards.
Books*
Book:John B. Conway/Functions of One Complex Variable/Second Edition
Book:Eberhard Freitag/Complex Analysis
Book:E.C. Titchmarsh/The Theory of Functions/Second Edition
Book:Richard K. Miller/Ordinary Differential Equations
Book:E.C. Titchmarsh/The Theory of Functions/Second Edition
Book:Charles C. Pugh/Real Mathematical Analysis/Second Edition
Book:Robert G. Bartle/Introduction to Real Analysis/Fourth Edition
Book:Brian S. Thomson/Symmetric Properties of Real Functions
Definitions*
Definition:Norm of Subdivision
Definition:Discontinuity of the First Kind
Definition:Real Interval/Bounded
Definition:Pointwise Equicontinuous
Definition:Uniformly Equicontinuous
Definition:Power (Algebra)#Real Numbers
Definition:Natural Logarithm/Positive Real/Definition 3
Definition:Less Than (Real Numbers)
Definition:Strictly Positive/Real Number/Definition 2
Definition:Real Exponential Function
Definition:Bounded Sequence/Complex/Unbounded
Definition:Bounded Sequence/Complex
Definition:Locally Bounded/Family of Mappings
Definition:Locally Bounded/Mapping
Definition:Euler's Number/Base of Exponential
Definition:Strictly Midpoint-Convex
Definition:Strictly Midpoint-Concave
Theorems
Set Theory*
Relative Difference between Infinite Set and Finite Set is Infinite
Mapping Theory*
Union of Functions Theorem/Corollary
Union of Inverses of Mappings is Inverse of Union of Mappings
Real Numbers/Order
Multiplication of Positive Number by Real Number Greater than One*
Power Function on Base Greater than One is Strictly Increasing/Real Number*
Rational Sequence Increasing to Real Number*
Rational Sequence Decreasing to Real Number*
Inequality of Product of Unequal Numbers*
Number Equal to Zero iff Arbitrarily Small*
Inequality iff Difference is Positive
Exponential is Monotonic/Rational Number*
Power of Positive Real Number is Positive/Natural Number
Power of Positive Real Number is Positive/Integer*
Power of Positive Real Number is Positive/Rational Number*
Power Function on Base between Zero and One is Strictly Decreasing/Natural Number*
Power Function on Base Greater than One is Strictly Increasing/Natural Number*
Power Function on Base between Zero and One is Strictly Decreasing/Integer*
Power Function on Base Greater than One is Strictly Increasing/Integer*
Power Function on Base between Zero and One is Strictly Decreasing/Rational Number*
Power Function on Base Greater than One is Strictly Increasing/Rational Number*
Root of Reciprocal is Reciprocal of Root*
Power Function is Strictly Increasing over Positive Reals/Natural Exponent*
Real Number between Zero and One is Greater than Power/Natural Number*
Power is Well-Defined/Rational
Transcendental Functions
Exponential Functions
Equivalence of Definitions of Real Exponential Function/Proof 2*
Exponential of Real Number is Strictly Positive/Proof 5/Lemma*
Exponential of Real Number is Strictly Positive/Proof 5*
Exponential of Real Number is Strictly Positive/Proof 4*
Exponential of Real Number is Strictly Positive/Proof 3*
Exponential of Real Number is Strictly Positive/Proof 2*
Exponential of Real Number is Strictly Positive/Proof 1*
Exponential of Product/Proof 3*
Exponential of Sum/Real Numbers/Proof 5*
Exponential of Sum/Real Numbers/Proof 4*
Power Function on Strictly Positive Base is Convex
Derivative of Exponential Function/Proof 5/Lemma*
Derivative of Exponential Function/Proof 5
Exponential Function is Well-Defined/Real*
Exponential Function is Well-Defined/Real/Proof 1*
Exponential Function is Well-Defined/Real/Proof 2
Exponential Function is Well-Defined/Real/Proof 3*
Exponential Function is Well-Defined/Real/Proof 4*
Exponential Function is Well-Defined/Real/Proof 5*
Exponent Combination Laws/Product of Powers/Proof 2/Lemma*
Derivative of Exponential Function/Proof 4*
Exponential Function is Continuous/Real Numbers/Proof 5*
Exponential Function is Well-Defined/Real/Proof 5*
Exponential Function is Well-Defined/Real/Proof 4/Lemma*
Exponent Combination Laws/Power of Power*
Exponent Combination Laws/Product of Powers*
Exponent Combination Laws/Product of Powers/Proof 2/Lemma*
Euler's Number to Rational Power permits Unique Continuous Extension*
Power Function to Rational Power permits Unique Continuous Extension
Power Function tends to One as Power tends to Zero/Rational Number
Power Function on Base between Zero and One Tends to One as Power Tends to Zero/Rational Number
Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number*
Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number/Lemma
Exponential Function is Continuous/Real Numbers/Proof 4*
Euler's Number: Limit of Sequence implies Base of Logarithm*
Exponential Sequence is Eventually Increasing*
Exponential Sequence is Eventually Strictly Positive
Derivative of Exponential Function/Complex*
Exponential Sequence is Uniformly Convergent on Compact Sets
Natural Logarithm
Real Natural Logarithm Function is Continuous/Proof 2
Defining Sequence of Natural Logarithm is Strictly Decreasing
Lower Bound of Natural Logarithm/Proof 3
Upper Bound of Natural Logarithm/Proof 2
Natural Logarithm of 1 is 0/Proof 3
Sum of Logarithms/Natural Logarithm/Proof 3
Logarithm of Power/Natural Logarithm/Natural Power
Logarithm of Power/Natural Logarithm/Integer Power
Logarithm of Power/Natural Logarithm/Rational Power
Defining Sequence of Natural Logarithm is Uniformly Convergent on Compact Sets
Derivative of Natural Logarithm Function/Proof 4/Lemma
Derivative of Natural Logarithm Function/Proof 4
Defining Sequence of Natural Logarithm is Strictly Decreasing
Lower Bound of Natural Logarithm/Proof 3
Upper Bound of Natural Logarithm/Proof 2
Logarithm of Power/Natural Logarithm/Rational Power
Logarithm of Power/Natural Logarithm/Integer Power
Logarithm of Power/Natural Logarithm/Natural Power
Sum of Logarithms/Natural Logarithm/Proof 3
Natural Logarithm of 1 is 0/Proof 3
Upper Bound of Natural Logarithm/Proof 2
Defining Sequence of Natural Logarithm is Convergent
Logarithm is Strictly Increasing and Strictly Concave/Corollary
Natural Logarithm as Derivative of Exponential at Zero
Real Analysis
Ordering of Series of Ordered Sequences
Product of Increasing Positive Functions is Increasing
Uniqueness of Continuously Differentiable Solution to Initial Value Problem
Discontinuity of Monotonic Function is Jump Discontinuity
Surjective Monotone Function is Continuous
Limit of Bounded Convergent Sequence is Bounded
Monotone Real Function with Everywhere Dense Image is Continuous
Finite Subset of Metric Space is Closed
Monotone Real Function with Everywhere Dense Image is Continuous/Lemma
Derivative of Uniformly Convergent Sequence of Differentiable Functions
Sum of Geometric Sequence/Corollary 2
Discontinuity of Monotonic Function is Jump Discontinuity
Surjective Monotone Function is Continuous
Limit of Bounded Convergent Sequence is Bounded
Subsets Inherit Uniform Convergence
Uniformly Convergent iff Difference Under Supremum Norm Vanishes
Pasting Lemma for Pair of Continuous Mappings on Closed Sets
Pasting Lemma for Pair of Continuous Mappings on Open Sets
Product of Uniformly Convergent Sequences of Bounded Functions is Uniformly Convergent
Uniformly Convergent Sequence of Bounded Functions is Uniformly Bounded
Uniformly Convergent Sequence Evaluated on Convergent Sequence
Continuous Midpoint-Convex Function is Convex
Continuous Strictly Midpoint-Convex Function is Strictly Convex
Continuous Midpoint-Concave Function is Concave
Continuous Strictly Midpoint-Concave Function is Strictly Concave
Power Function on Base Greater than One is Strictly Convex