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
Author(s): The Open University

You have seen that the complex number x + iy corresponds to the point (x, y) in the complex plane. This correspondence enables us to give an alternative description of complex numbers, using so-called polar form. This form is particularly useful when we discuss properties related to multiplication and division of complex numbers.

Author(s): The Open University

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

A little history is instructive here, because it shows how difficult Theorem 9 really is. By 1900 the classification of compact surfaces was well understood, although proofs of the major theorems relied more on intuition than would be acceptable today. Attention switched to objects called â€˜3-manifoldsâ€™, topological s
Author(s): The Open University

1. By â€˜containsâ€™, we mean that we can find part of the surface that is homeomorphic to a MÃ¶bius band. The edge of the MÃ¶bius 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 MÃ¶bius 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
Author(s): The Open University

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 â€“
Author(s): The Open University

Understanding Cities
A short introduction to this album.
Author(s): No creator set

Lesson 01 - One Minute Luxembourgish
In lesson 01 of One Minute Luxembourgish you will learn how to say 'hello' and 'goodbye' in Luxembourgish. Remember - even a few phrases of a language can help you make friends and enjoy travel more. Find out more about One Minute Languages at our website - http://www.oneminutelanguages.com. One Minute Luxembourgish is brought to you by the Radio Lingua Network and is Â©Copyright 2008.Author(s): No creator set

Volcanic Hazards in New Zealand
Brad Scott describes the different volcano types and potential threats in New Zealand. This includes active volcano footage thanks to Geoff Mackley.  (04:44)
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Earthquake Probabilities and Forecasting
Scientist, Matt Gerstenberger, uses statistical models to forecast aftershock probabilities of earthquakes.  He explains his work and why it is important.  (05:28)
Author(s): No creator set

Engaging with Detroit
University of Michigan students live, study and work in the Motor City
Author(s): No creator set

Jordan Peterson on Redemption and Psychology in Christianity
University of Toronto Psychology Professor, Jordan Peterson, on Redemption and Psychology in Christianity. Dr. Peterson's lecture was the keynote address at the 2012 Meaning Conference held in Toronto.
Author(s): No creator set

Opus
Animation from COFA student Jessica McGregor. Winner of best 3D CGI animation at the 2012 COFA Annual Awards.
Author(s): No creator set

Topic 6: Social Insurance Part 3 | Econ2450A: Public Economics
Raj Chetty Fall 2012
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History of the Incas
General introduction to the history of the Inca people and their influence, which extended through an enormous area of Latin America.
Author(s): No creator set

Course Organizational Plan
Week Three, OHCC
Author(s): No creator set

Human Impact on the Biosphere

Author(s): No creator set

Windows PowerShell v3 is here, and it is delivering on Microsoft's promise to make nearly everything in Windows manageable from the command-line. Are you finally going to learn the shell, or learn to say, "would you like fries with that?" instead? Join PowerShell author, columnist, trainer, and MVP Don Jones (one of the world's most well-known PowerShell experts) in a crash course that shows you how to use the shell's key features. No scripting experience neededâ€”you'll use the she
Author(s): Don Jones, Jeffrey Snover

Tonkatsu Sliders
Chef Jet Tila from Charleston Gastropub demonstrates how to make sliders filled with tonkatsu, tonkastu sauce, shredded cabbage and sriracha mayonnaise. To make the tonkatsu, he dredges a pork shoulder steak with kara-age and panko, then fries the pork until golden brown. He stuffs bao, or Chinese buns, with the tonkatsu and adds tonkastu sauce balanced with creamy and spicy sriracha mayonnaise. This video series was created by the Culinary Institute of America as an industry service to Kikkoman
Author(s): No creator set

Hungarian scientists discover remains of first freshwater 'mosasaur', an ancient lizard
Dec. 27 - Hungarian scientists discover ancient fossils of a new giant lizard species that lived in freshwater. A reconstruction of the 'mosasaur' has gone on display in the Hungarian Natural History Museum as the palaeontologists published their findings. Joel Flynn reports.
Author(s): No creator set