User:Caliburn/Surgery

These are all the errors or gaps in my work that I'm aware of, they are mainly singular dubious steps concerning interchanging limits (sorry - wrote these some time ago, I am surprised so much is correct)

• Parseval's Theorem/Formulation 2 - interchange of sum and integrals needs to be tightened up
• Riemann Zeta Function as a Multiple Integral - interchange
• Sum of Reciprocals of Squares of Odd Integers/Proof 2 - interchange.
• Sum of Reciprocals of Squares of Odd Integers/Proof 3 - maybe be able to rescued
• Derivative of Arctangent Function/Proof 2 - justify Big-$\OO$ properly
• Laurent Series Expansion for Cotangent Function - is swap of sums fine?
• Mittag-Leffler Expansion for Cotangent Function/Proof 1 - serious technical gaps
• Definition:Spence's Function and related - notational and technical gaps
• Cauchy's Residue Theorem - needs tightening
• Argument Principle - pretty sure this is technically up to snuff but concept logarithmic derivative absent
• Sum of Reciprocals of Powers of Odd Integers Alternating in Sign - use Integral of Series of Positive Measurable Functions
• Definite Integral from 0 to 1 of Power of u over 1 + Power of u - the interchange of integral and sum needs to be tightened
• Summation Formula (Complex Analysis)/Lemma - could use a bit of extra explanation
• Integral Representation of Dirichlet Beta Function in terms of Gamma Function - reference to DCT needs checking
• Definite Integral from 0 to 1 of x to the x - interchange needs justifying
• Definite Integral from 0 to 1 of x to the minus x - interchange needs justifying
• Definite Integral to Infinity of Logarithm of Exponential of x plus One over Exponential of x minus One - ditto
• Integral Representation of Dirichlet Eta Function in terms of Gamma Function - interchange
• Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x - interchange
• Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One plus x - interchange
• Definite Integral to Infinity of Sine of a x over Hyperbolic Sine of b x - interchange
• Definite Integral to Infinity of Hyperbolic Sine of a x over Exponential of b x minus One - interchange

Looking into it, it seems I'm daft. Apparently a lot them are "really" Fubini's with one measure the counting measure. Will need separate proof but that means a lot of this (except possibly when dealing with series that aren't absolutely convergent, which will need a bit more work) isn't really "broken" just under-justified.