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...2) is differentiable at every point of double-struck cap r minus zero . This reflects the fact that complex differentiation imposes a much stronger condition on functions than does real differentiation. Indeed, as the course progresses, you will see that differentiable complex functions have remarkably pleasant properties. For example, if a complex function can be...

...2 nm in diameter. Because of their small size, they have a large surface area and physical and chemical properties which can easily be manipulated. The unique properties of the gold surface mean we can attach different therapeutic molecules (such as drugs, DNA and antibodies) to their surface. As gold nanoparticles can bear a high drug load, this makes them ideal carriers...

...2 flyby missions began sending images back to Earth, there was very little evidence of impacts. Instead, Pioneer 10 showed the ice was largely smooth and flat, with dark red and light patches, and Voyager 1 showed a surface covered in intersecting dark red lines (now known as lineae) (Figure 2). [Photograph of Europa taken by Voyager 2 - it shows approximately half of...

...2 introduces a scalar field, called the stream function, which for an incompressible fluid provides an alternative method of modelling the flow and finding the streamlines. Sections 2 and 3 derive the stream functions for several simple two-dimensional flow types (the uniform flow, source, doublet and vortex), and suitable combinations of these are used to model...

...2). This is a typical optimisation problem which can be used to demonstrate the power of quantum computing as well as being a problem which delivery companies would like to be able to solve quickly. [Described image] Figure 2 An illustration of the travelling salesperson problem. The image on the left shows 7 cities (in blue) and a trial of all 360 possible routes between...