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2.10 The complex exponential function

Consider the real exponential function f (x) = ex (that is, f (x) = exp x). We now extend the definition of this function to define a function f(z) = ez whose domain and codomain are .

We expect complex powers
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2.1 What is a complex number?

We will now discuss complex numbers and their properties. We will show how they can be represented as points in the plane and state the Fundamental Theorem of Algebra: that any polynomial equation with complex coefficients has a solution which is a complex number. We will also define the function exp of a complex variable.

Earlier we mentioned several sets of numbers, including Author(s): The Open University

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3.2.1 Remarks

  1. By ‘contains’, we mean that we can find part of the surface that is homeomorphic to a Möbius band. The edge of the Möbius band does not need to correspond to an edge at the surface, so that a surface without boundary can be non-orientable (as we shall shortly see).

  2. When seeking Möbius bands in a surface, it can be helpful to look at all possible closed curves on the surface and thicken these into bands.

  3. Remember, fro
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2.4.1 Remarks

  1. This theorem applies to all surfaces and not just to surfaces in space.

  2. This theorem tells us that the boundary number is a topological invariant for surfaces, i.e. a property that is invariant under homeomorphisms.

  3. It follows from the theorem that two surfaces with different boundary numbers cannot be homeomorphic. It does not follow that two surfaces with the same boundary number are homeomorphic –
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2.1 Surfaces in space

In Section 2 we start by introducing surfaces informally, considering several familiar examples such as the sphere, cube and Möbius band. We also illustrate how surfaces can be constructed from a polygon by identifying edges. A more formal approach to surfaces is presented at the end of the section.

Figure 3 shows
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Studying mammals: Return to the water
Some of the most unusual and versatile of all the mammals are the groups that live, feed and reproduce underwater. In this unit we will see how these formerly land-based mammals adapted to a return to the water, discussing such challenges as breathing, movement and communication. This is the seventh unit in the ‘Studying mammals’ series. First pu
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Identity and Access Control

Today I woke up thinking that talking about Identity and Access Control and how your strategy around that affects you (web-) app's architecture without going too deeply into the security lingo that usually comes with it. Here's the 40 minute result.

I start with HTTP's "native" authentication model RFC 2617 and how that's universally bad, with both Basic and Digest authentication having issues Digest being, ironically worse for the overall security strategy. Then I d
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