# 数学代写|复分析作业代写Complex function代考|The Fundamental Theorem of Contour Integration

#### Doug I. Jones

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## 数学代写|复分析作业代写Complex function代考|The Fundamental Theorem of Contour Integration

Integrals of complex functions are only occasionally computed by breaking them down into real and imaginary parts and calculating two real integrals as in the previous section. This technique is sometimes needed, but a far more efficient method is available for $\int_\gamma f$ if we can find an antiderivative of $f$. This concept is so important that we give it a formal definition:

DEFINITION 6.32. Let $D$ be a domain. An antiderivative for a function $f: D \rightarrow \mathbb{C}$ is a function $F: D \rightarrow \mathbb{C}$ such that $F^{\prime}=f$.

An antiderivative, if it exists, is unique up to an added constant, for if $F^{\prime}=G^{\prime}=f$ in a domain, then $(F-G)^{\prime}=0$ so $F-G$ is constant by Theorem 4.14. If we can find an antiderivative, then the integral can be computed immediately using:

TheOREm 6.33 (Fundamental Theorem of Contour Integration). If $f: D \rightarrow \mathbb{C}$ is continuous, $F: D \rightarrow \mathbb{C}$ is an antiderivative of $f$, and $\gamma$ is a contour in $D$ from $z_0$ to $z_1$, then
$$\int_\gamma f=F\left(z_1\right)-F\left(z_0\right)$$
Proof. Let $w(t)=u(t)+\mathrm{i} v(t), W(t)=U(t)+\mathrm{i} V(t)$ for $t \in[a, b]$, where $u, v, U, V$ are real. Then $W^{\prime}=w$ if and only if $U^{\prime}=u, V^{\prime}=v$, and then
\begin{aligned} \int_a^b w(t) \mathrm{d} t & =\int_a^b u(t) \mathrm{d} t+\mathrm{i} \int_a^b v(t) \mathrm{d} t \ & =U(b)-U(a)+\mathrm{i} V(b)-\mathrm{i} V(a) \ & =W(b)-W(a) \end{aligned}
Let $w(t)=f(\gamma(t)) \gamma^{\prime}(t)$. Since $F^{\prime}=f$,
$$w^{\prime}(t)=F^{\prime}(\gamma(t)) \gamma^{\prime}(t)=W^{\prime}(t)$$
where $W(t)=F(\gamma(t))$. Therefore
\begin{aligned} \int_\gamma f & =\int_a^b w(t) \mathrm{d} t=W(b)-W(a) \ & =F(\gamma(b))-F(\gamma(a))=F\left(z_1\right)-F\left(z_0\right) \end{aligned}

## 数学代写|复分析作业代写Complex function代考|The Gamma Function

So far the special complex functions that we have encountered are all extensions to $\mathbb{C}$ of familiar real functions: polynomials, powers, trigonometric functions, the exponential and the logarithm. In this section we briefly discuss a less familiar function, the gamma function $\Gamma(z)$. This also began as a real function and was later extended to the complex case. It has important applications, but we restrict attention to defining the function and establishing a few basic properties. It will play a key role in Chapter 17 when we sketch connections between complex analysis and number theory, culminating in the Riemann Hypothesis.

The definition of the gamma function goes back to Euler, who initially defined it as the infinite product
$$\Gamma(z)=\frac{1}{z} \prod_{n=1}^{\infty}\left[\left(1+\frac{1}{n}\right)^z\left(1+\frac{z}{n}\right)^{-1}\right]$$
This infinite product converges for all $z \in \mathbb{C} \backslash{0,-1,-2, \ldots}$. It defines a differentiable function on that domain, but we lack the techniques needed to prove this. Instead, we use another formula due to Euler, who showed that when re $z>0$ this is equal to the integral
$$\Gamma(z)=\int_0^{\infty} t^{z-1} \mathrm{e}^{-t} \mathrm{~d} t$$
We take this as our definition, in order to illustrate complex integration. Here we define
$$\int_0^{\infty} f=\lim {a \rightarrow \infty} \int{[0, a]} f$$
so the interval $[0, \infty]$ is interpreted as the non-negative real axis. It is straightforward to verify that (6.17) converges, because the factor $\mathrm{e}^{-t}$ tends to zero rapidly. With this definition, the formula (6.17) defines $\Gamma(z)$ only for $z$ in the positive half-plane

$$\mathbb{C}^{+}={z \in \mathbb{C}: \text { re } z>0}$$
We establish some basic properties of $\Gamma(z)$ using methods from complex integration.

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|The Fundamental Theorem of Contour Integration

6.32.定义让$D$成为一个域名。一个函数$f: D \rightarrow \mathbb{C}$的不定积分是一个函数$F: D \rightarrow \mathbb{C}$使得$F^{\prime}=f$。

$$\int_\gamma f=F\left(z_1\right)-F\left(z_0\right)$$

\begin{aligned} \int_a^b w(t) \mathrm{d} t & =\int_a^b u(t) \mathrm{d} t+\mathrm{i} \int_a^b v(t) \mathrm{d} t \ & =U(b)-U(a)+\mathrm{i} V(b)-\mathrm{i} V(a) \ & =W(b)-W(a) \end{aligned}

$$w^{\prime}(t)=F^{\prime}(\gamma(t)) \gamma^{\prime}(t)=W^{\prime}(t)$$

\begin{aligned} \int_\gamma f & =\int_a^b w(t) \mathrm{d} t=W(b)-W(a) \ & =F(\gamma(b))-F(\gamma(a))=F\left(z_1\right)-F\left(z_0\right) \end{aligned}

## 数学代写|复分析作业代写Complex function代考|The Gamma Function

$$\Gamma(z)=\frac{1}{z} \prod_{n=1}^{\infty}\left[\left(1+\frac{1}{n}\right)^z\left(1+\frac{z}{n}\right)^{-1}\right]$$

$$\Gamma(z)=\int_0^{\infty} t^{z-1} \mathrm{e}^{-t} \mathrm{~d} t$$

$$\int_0^{\infty} f=\lim {a \rightarrow \infty} \int{[0, a]} f$$

$$\mathbb{C}^{+}={z \in \mathbb{C}: \text { re } z>0}$$

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