# Category:Infinite Products

Jump to navigation
Jump to search

This category contains results about Infinite Products.

Definitions specific to this category can be found in Definitions/Infinite Products.

Let an infinite number of values of $j$ satisfy the propositional function $R \left({j}\right)$.

Then the precise meaning of $\displaystyle \prod_{R \left({j}\right)} a_j$ is:

- $\displaystyle \prod_{R \left({j}\right)} a_j = \left({\lim_{n \mathop \to \infty} \prod_{\substack {R \left({j}\right) \\ -n \mathop \le j \mathop < 0} } a_j}\right) \times \left({\lim_{n \mathop \to \infty} \prod_{\substack {R \left({j}\right) \\ 0 \mathop \le j \mathop \le n} } a_j}\right)$

provided that both limits exist.

If either limit *does* fail to exist, then the **infinite product** does not exist.

## Subcategories

This category has the following 5 subcategories, out of 5 total.

## Pages in category "Infinite Products"

The following 47 pages are in this category, out of 47 total.

### A

- Absolute Value of Absolutely Convergent Product is Absolutely Convergent
- Absolute Value of Convergent Infinite Product
- Absolute Value of Divergent Infinite Product
- Absolute Value of Infinite Product
- Absolute Value of Uniformly Convergent Product
- Absolutely Convergent Product Does not Diverge to Zero
- Absolutely Convergent Product is Convergent

### D

### E

### F

### L

- Logarithm of Convergent Product of Real Numbers
- Logarithm of Divergent Product of Real Numbers
- Logarithm of Infinite Product of Complex Functions
- Logarithm of Infinite Product of Complex Functions/Corollary
- Logarithm of Infinite Product of Complex Numbers
- Logarithm of Infinite Product of Real Numbers
- Logarithmic Derivative of Infinite Product
- Logarithmic Derivative of Infinite Product of Analytic Functions
- Logarithmic Derivative of Infinite Product of Holomorphic Functions

### P

### U

- Uniform Absolute Convergence of Infinite Product of Complex Functions
- Uniform Product of Analytic Functions is Analytic
- Uniform Product of Continuous Functions is Continuous
- Uniformly Absolutely Convergent Product is Uniformly Convergent
- Uniformly Convergent Product Satisfies Uniform Cauchy Criterion