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From that time onwards it would be assumed that analytic function theory, which had already unified the trigonometric and exponential functions, was a fundamental tool.
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Conversely, given any harmonic function in two dimensions, it is the real part of an analytic function, at least locally.
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However, the square of the distance (denoted or), which has a paraboloid as its graph, is a smooth and analytic function.
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The initial impetus was to have a description of the properties of the domain of definition of the (maximal) analytic continuation of an analytic function.
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One can easily prove that any analytic function of a real argument is smooth.
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In both cases "f" is an analytic function on the corresponding open half plane.
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They follow directly from the fact that the order of differentiation of an analytic function of two variables is irrelevant.
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This leads to bounds on the number of zeros of an analytic function in a complex region.
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The concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function.
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Then the period of those forms is an analytic function which splits into a product of local factors.
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