1.2 Calculus
Since traditional complex analysis is the theory of calculus on complex functions, it is only natural that generalizations are made on classical formulas in calculus for complex functions.
It is well known that a function is differentiable at a point if the limit
exists, and the value of this limit is the derivative of , denoted by or . The value is the differential of . Partition into such that the length of the intervals vanishes (we let the norm of the partition, or the size of the largest interval, tend to zero) as . If for any such partition, the sum
tends to the same value (as the length of the largest partition approaches 0), then the function can be roughly said to be integrable over . The full details of Riemann integrability are simplified by the use of Darboux sums and will not be discussed here. The value of this sum is denoted by
We will attempt to avoid notions involving Lebesgue integration. However, it is important to note that every Riemann integrable function is also Lebesgue integrable, and the two integrals are equal. Therefore, we will use Lebesgue integral theorems (where the resultant integral is Riemann integrable) when necessary without further mention of the Lebesgue integral itself.
The following theorems are the fundamental results of classical calculus:
Theorem 1.2.1 (Fundamental Theorem of Calculus, Differential Form).
Let be a function continuous over . For , define
Then is differentiable over , , and .
Theorem 1.2.2 (Fundamental Theorem of Calculus, Integral Form).
Let be a function differentiable over . Let over . Then,
The two forms of the theorem show that differentiation and integration are inverse operations to each other. Operations performed for differentiating oftentimes have a corresponding inverse operation that can be done for integrating. For instance,
corresponds to
and
corresponds to
and
corresponds to
Another correspondence is the Mean Value Theorem:
Theorem 1.2.3 (Mean Value Theorem, Differential Form).
If is differentiable over , then such that
Theorem 1.2.4 (Mean Value Theorem, Integral Form).
If is continuous over , then such that
A curve is a one-dimensional manifold embedded within a higher dimensional space. They can be parameterized with a vector of one parameter. In the complex plane, a curve is a complex-valued function for a real parameter . A curve is closed if . It is smooth if it is continuously differentiable, and its direction is defined to be the direction as increases. If it is smooth everywhere except at a finite number of points, it is piecewise smooth. If it is of finite length, then the curve is said to be rectifiable. Piecewise smooth curves are rectifiable. A curve is simple if it is simple (non-self-intersecting), or if implies that . A simple closed curve is also called a Jordan curve.
Theorem 1.2.5 (Jordan Curve Theorem).
Let be a Jordan curve in . Then the set consists of exactly two connected subsets. One of them is the interior, denoted by , and is a bounded set, while the other is the exterior, denoted by , which is unbounded. Both of the two sets share the common boundary .
The theorem above seems trivial, but its rigorous proof in topology is extremely complex. The theorem itself can also be stated on instead of . For a region , the boundary is denoted . If the region bounded by any closed curve in also lies in , then it is a simply connected region. A connected region that is not simply connected is multiply connected. A region bound by 2 non-intersecting Jordan curves is doubly connected, and a region bound by non-intersecting Jordan curves is traditionally known as -connected. Lastly, any closed curve can degenerate to a single point or slit.
Generalizations of the differential and integral exist for multivariate functions. The partial differentials of , , , and sum up to form the total differential, denoted by . An important result in multivariable calculus allows the calculation of the derivatives of a definite integral with respect to its parameter.
Theorem 1.2.6 (Leibniz Integral Rule).
Let be continuous on , , and suppose are differentiable functions of . If is continuously differentiable with respect to , then
Four main classical theorems exist, relating a function and its line integral in 2 and 3 dimensions, line and surface (or area) integrals in 2 and 3 dimensions, and the surface and volume integrals in 3 dimensions:
Theorem 1.2.7 (Gradient Theorem).
Let be an oriented smooth curve in with boundary points and . Then if
Theorem 1.2.8 (Green's Theorem).
Let be a positively oriented, multiply connected subset of with a piecewise smooth oriented boundary . Suppose that . Then,
Theorem 1.2.9 (Stokes' Theorem).
Suppose that is a positively oriented surface with a positively oriented, piecewise smooth boundary curve . Suppose that . Then,
Theorem 1.2.10 (Gauss' Theorem).
Suppose that is a positively oriented region with a positively oriented, piecewise smooth boundary surface . Suppose that . Then,
In 3-dimensional space, define a scalar valued function to be a 0-form, a linear combination of , , and to be a 1-form, and a linear combination of , , and to be a 2-form, and to be a 3-form, where denotes an anti-commutative and associative product, where for any two differential forms and
Then consequently, for any differential form ,
We can generalize the operator to increase the degree of a differential form. For instance,
which is the definition of the total differential. For a 1-form in 3-dimensional space, , we can define the exterior derivative in a similar way:
Similarly, we can differentiate a 2-form to get:
The two results above resemble the curl and divergence of . A differential form is closed if , and is exact if there exists such that .
Lemma 1.2.1 (Poincaré).
For any differential form on an open, contractible set , if is closed, then it is also exact.
It is true that for any set , regardless of contractibility, that for a differential form defined on , . In other words, all exact differential forms are closed. (For a region , we have . This is one of many reasons for which the boundary operator is denoted by , in analogy to .)
The implications of this are important: if is a 0-form, then , and if is a 1-form, , where is the vector of the coefficients of the basis differential forms of (there are no correlations for higher degree forms since in 3-dimensional space, the highest degree possible for any differential form is 3).
Then, the Fundamental Theorem of Calculus, the Gradient Theorem, Green’s, Stokes’, and Gauss’ Theorems can be generalized into:
Theorem 1.2.11 (Stokes–Cartan).
For an oriented smooth -dimensional compact manifold with boundary , for a smooth differential -form over ,
Real analysis is the subject dedicated to rigorously defining concepts such as limits, continuity, integrability, convergence, etc. The most widely used definition of a finite limit of a function is the language of –, which states:
Definition 1.2.1 (Epsilon–Delta).
Let be a function defined over an open set such that is an accumulation point of . We say that if , such that for all with , we have .
Similarly, we define the right-handed limit if for every , there exists such that for all with , we have .
Likewise, the left-hand limit exists if for every , there exists such that for all with , we have .
We also have the definition of the limit of a sequence:
Definition 1.2.2 (Epsilon–N).
Let be a sequence. If such that , such that , , then converges to .
Theorem 1.2.12 (Cauchy Criterion).
Let be a sequence. Then is convergent iff , such that , .
Proof.
Assume is convergent. Then , such that , and for some . It follows that
Conversely, is bounded (fixing , , ). By the Bolzano–Weierstrass Theorem (Theorem 1.1.2), has a subsequence that converges to . Therefore, , and such that , , and , and . Then
Hence, converges to .
Definition 1.2.3 (Limit Superior).
For a number sequence , if such that:
- , such that , ,
- , , such that ,
then the superior limit of is , denoted by .
Definition 1.2.4 (Limit Inferior).
For a number sequence , if such that:
- , such that , ,
- , , such that ,
then the inferior limit of is , denoted by .
Lemma 1.2.2.
A number sequence is convergent iff .
Proof.
We first prove that implies . By Definition 1.2.2, , such that ,
Then from Definition 1.2.3 and Definition 1.2.4, we have that and . By the second conditions, we get and . Therefore,
For the converse, assume . Since such that , . such that , . Then , , as expected.
Definition 1.2.5 (Continuity).
A function , defined on an open set containing a point , is said to be continuous at iff
It is important to note that in the case of multiple explicit variables, a distinction is made between (separate) continuity (where there are two ‘s on which variable varies, and does not guarantee a single for when both variables vary simultaneously) and joint continuity (where a single controls both variables at once). To illustrate this, let be fixed. The former is commonly written as
in conjunction with
whereas the latter is expressed as
Theorem 1.2.13.
Any continuous function on a compact set is bounded on .
Proof.
Suppose for the sake of contradiction that is continuous and unbounded on compact . Then for each , there exists such that . The sequence lies in , which is compact, so by the Bolzano–Weierstrass Theorem (Theorem 1.1.2), has an accumulation point in . In other words, there exists a convergent subsequence with .
Since is continuous, , which is well-defined because . However, this contradicts , hence must be bounded on .
Theorem 1.2.14 (Extreme Value).
A continuous function defined on a compact set attains its infimum and supremum in .
Proof.
Assume that never attains its supremum . Then, . Define the auxiliary function , which is strictly positive and continuous as the denominator never reaches . By Theorem 1.2.13, is bounded with some value of satisfying . also has the representation , and therefore,
which means that is not the supremum. Similarly, assume that never attains its infimum . Then . Let , which is strictly positive and continuous as the denominator never reaches . By Theorem 1.2.13, is bounded with some value of satisfying . also has the representation , and therefore,
which means that is not the infimum.
Definition 1.2.6 (Uniform Continuity).
A function , defined on a set , is uniformly continuous iff , such that where , .
Example 1.2.1.
The function is not uniformly continuous over .
Proof.
If such that , satisfying both and , then is not uniformly continuous over .
Let and
Then , where , since . Additionally, . This satisfies the negation, and thus, is not uniformly continuous over .
Theorem 1.2.15 (Heine–Cantor).
A continuous function on a compact set is uniformly continuous on .
Proof.
Fix . Since is continuous at , for every there exists such that for all ,
The collection of open balls forms an open cover of the compact set . By Heine–Borel (Theorem 1.1.3), there is a finite subcover
Set
Now let satisfy . Then there exists an index such that . Consequently,
Applying (1.2.1) to the points and through , we obtain
Therefore,
Since was arbitrary, the uniform continuity of on follows.
Definition 1.2.7.
A function is Lipschitz continuous over if such that , . The smallest possible satisfying the above condition is known as the Lipschitz constant.
Lipschitz continuity is an important concept in real analysis and the theory of differential equations. It is a strong form of uniform continuity.
Proposition 1.2.1.
All Lipschitz continuous functions on are uniformly continuous on .
Proof.
Let be the Lipschitz constant. Then , let . It then follows that such that , .
Proposition 1.2.2.
A function on a compact set is Lipschitz continuous on .
Proof.
Let be . By Theorem 1.2.13, since is compact and is continuous, such that , .
By the Mean Value Theorem, , between and such that . Then, , which means is Lipschitz continuous with Lipschitz constant less than or equal to .