By Frederick S. Woods
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Additional info for A course in mathematics,: For students of engineering and applied science,
Deﬁnitions like this one can be hard to take in, because they involve holding in one’s mind three diﬀerent levels of complexity. At the bottom we have real numbers, denoted by x and y. In the middle are functions like f , u, and T f , which turn real numbers (or pairs of them) into real numbers. At the top is another function, T , but the “objects” that it transforms are themselves functions: it turns a function like f into a diﬀerent function T f . This is just one example where it is important to think of a function as a single, elementary “thing” rather than as a process of transformation.
Then the linear approximation to f near z has the matrix ⎛ ⎞ ∂u ∂u ⎜ ∂x ∂y ⎟ ⎜ ⎟ ⎜ ⎟. ⎝ ∂v ∂v ⎠ ∂x ∂y The matrix of an expansion and rotation always has the a b form ( −b a ), from which we deduce that ∂u ∂v = ∂x ∂y and ∂u ∂v =− . ∂y ∂x These are the Cauchy–Riemann equations. One consequence of these equations is that ∂2u ∂2u ∂2v ∂2v + = − = 0. 4). A similar argument shows that v does as well. These facts begin to suggest that complex diﬀerentiability is a much stronger condition than real diﬀerentiability and that we should expect holomorphic functions to have interesting properties.
Be a sequence of real numbers. What does it mean to say that these numbers approach a speciﬁed real number l? The following two examples are worth bearing in mind. The ﬁrst is the sequence 12 , 32 , 34 , 45 , . . In a sense, the numbers in this sequence approach 2, since each one is closer to 2 than the one before, but it is clear that this is not what we mean. What matters is not so much that we get closer and closer, but that we get arbitrarily close, and the only number that is approached in this stronger sense is the obvious “limit,” 1.