Glasser主定理的一个推广及一批常涉及Riemann zeta函数的反常积分
An Extension of Glasser's Master Theorem and a Collection of Improper Integrals Many of Which Involve Riemann's Zeta Function
AI总结：
本文通过引入两个新参数推广Glasser主定理，并利用该推广定理大量求值涉及Riemann zeta函数的反常积分，展示其广泛适用性。
摘要：
Glasser主定理（arXiv:1308.6361v2）本质上是Cauchy积分定理被化归为一种专门形式后的重述。本文通过引入两个新参数对该定理进行推广，同时仍保持其形式简洁。鉴于人们对涉及Riemann zeta函数的对象有广泛兴趣，本文重点研究被积函数几乎任意地包含该函数的反常积分的求值，但也考虑其他一些情形，目的是展示推广后定理的威力。这一目标通过给出大量例子来实现，这些例子说明了可能取值范围的普遍性。本研究的一个简单成果是利用推广后的定理表明：当一个被积函数的级数表示或其他表示不收敛时，如何仍有可能求出相应积分。
英文摘要：
Glasser's Master Theorem arXiv:1308.6361v2 is essentially a restatement of Cauchy's integral Theorem reduced to a specialized form. Here we extend that theorem by introducing two new parameters, but still retain a simple form. Because of wide interest in entities involving Riemann's zeta function, the focus is on the evaluation of improper integrals with almost arbitrary integrands involving that function, but we also consider some other instances, the purpose being to demonstrate the power of the extended theorem. This is achieved by the presentation of a large number of examples that illustrate the ubiquity of the range of possibilities. One simple outcome of the study is the use of the extended theorem to show how it is possible to evaluate an integral when series or other representations of an integrand function do not converge.
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