Mathematical functions - Mathematical functions From...

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Mathematical functions From Physics Notes This page presents a brief summary of common mathematical functions and their properties. Students of MH2801 Complex Methods for the Sciences should already be familiar with most of the concepts presented here. Contents 1 Standard mathematical functions 1.1 Exponential and logarithm 1.2 Trigonometric functions 1.3 Hyperbolic functions 2 Continuity 3 Exercises Standard mathematical functions A mathematical function, denoted , takes an input (which is sometimes also called an argument ), and returns an output . For now, we consider the case where both and are real numbers. The set of possible inputs is called the domain of the function, and the set of possible outputs is called the range . A well-defined function must have an unambiguous output: for any in the domain, must be a specific number in the range. In other words, functions must be either one-to-one ("injective") mappings, or many-to-one mappings. They can't be one-to-many or many-to-many. This is illustrated in Fig. 1. Fig. 1: Graphs of one-to-one, many-to-one, and one-to-many mappings. The first two are well- defined functions, but the last one is not. Simple examples of mathematical functions are those based on elementary algebra operations: Exponential and logarithm The exponential function is a particularly important and ubiquitous function. You've probably come across this function before, but let's remind ourselves of how and why it's defined. We begin with a meditation on what it means to take a number to the power of : For values of in natural numbers , the power operation simply means multiplying by itself times. For example, . But what about non natural number powers, like or or ? To answer this question, we introduce the natural exponential function, defined as the following limiting infinite series:
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Fig. 2: Graphs of the exponential function , and the natural logarithm function . Fig. 3: The basic trigonometric functions, defined using a right-angled triangle with a unit hypothenuse. The angle is equal to the length of the circular arc joining the horizontal axis to the tip of the hypothenuse.
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