Complex differentiability
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Let us now define what complex differentiability is.
- Example 2.3.2
The function
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is nowhere complex differentiable.
- Proof
Let be arbitrary. Assume that is complex differentiable at , i.e. that
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exists.
We choose
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Due to lemma 2.2.3, which is applicable since of course is open, we have:
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But
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a contradiction.
The Cauchy–Riemann equations
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We can define a natural bijective function from to as follows:
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In fact, is a vector space isomorphism between and .
The inverse of is given by
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Theorem and definitions 2.3.3:
Let be open, let be a function and let . If is complex differentiable at , then the functions
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are well-defined, differentiable at and satisfy the equations
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These equations are called the Cauchy-Riemann equations.
- Proof
1. We prove well-definedness of .
Let . We apply the inverse function on both sides to obtain:
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where the last equality holds since is bijective (for any bijective we have if ; see exercise 1).
3. We prove differentiability of and and the Cauchy-Riemann equations.
We define
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Then we have:
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From these equations follows the existence of , since for example
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exists due to lemma 2.2.3.
The proof for
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and the existence of we leave for exercise 2.
Holomorphic functions
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- Let be sets such that , and let be a bijective function. Prove that .
- Let be open, let be a function and let . Prove that if is complex differentiable at , then and exist and satisfy the equation .
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