Goursat's Theorem and Cauchy's Theorem in a Convex Domain — Examples

Given: The oriented triangle $\mathrm{\Delta }[0,1,i]$ and the integrand $f(z)=z^2$.

The oriented triangle boundary follows the directed edges $0\to 1\to i\to 0$ (Filled complex triangles, their oriented three-edge boundary contours, diameter, and perimeter).

Complex polynomials are entire and obey the power derivative rule (Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero).

If $F$ is holomorphic with continuous derivative on an open set containing a rectifiable contour from $u$ to $v$, then $\int F^{\mathrm{\prime }}=F(v)-F(u)$ (The line integral of a continuous function admitting a primitive is that primitive's endpoint increment along every rectifiable path).

Goursat's theorem gives zero integral for a holomorphic function around every filled triangle contained in its domain (Goursat's triangle theorem: a holomorphic function integrates to zero around every triangle contained in its domain).

A holomorphic function is continuous (Complex differentiability at a point implies continuity there).

Given: The displayed vertices and midpoints, with every triangle carrying the orientation fixed by its ordered vertices.

If $p,q,r$ are the side midpoints of $\mathrm{\Delta }[a,b,c]$ and $f$ is continuous on that filled triangle, then $I_f[a,b,c]=I_f[a,p,r]+I_f[p,b,q]+I_f[r,q,c]+I_f[p,q,r]$ (Midpoint subdivision of a triangle cancels every interior edge and preserves its outer boundary integral).

Given: The positively oriented radius-$2$ circle $\gamma$ and the integrand $e^z\mathrm{/}(z-1)$.

The complex exponential is entire (The complex exponential is entire and its complex derivative is itself).

If $f$ is holomorphic on $D(a,R)$, $0<r<R$, $\mathrm{\mid }z-a\mathrm{\mid }<r$, and $\gamma (t)=a+r\exp (it)$ for $0\le t\le 2\pi$, then ${\int }_{\gamma }f(\zeta )\mathrm{/}(\zeta -z)d\zeta =2\pi if(z)$ (Cauchy's integral formula on a circle compactly contained in a disc of holomorphy).

Given: The positively oriented radius-$2$ circle and the displayed integrand.

If $f$ is holomorphic on $D(a,R)$, $0<r<R$, $z\in D(a,r)$, $n\in \mathbb{N}$, and $\gamma (t)=a+r\exp (it)$ for $0\le t\le 2\pi$, then ${\int }_{\gamma }f(\zeta )\mathrm{/}(\zeta -z{)}^{n+1}d\zeta =(2\pi i\mathrm{/}n!)f^{(n)}(z)$ (All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle).

The complex cosine is entire, with $(\cos z{)}^{\mathrm{\prime }}=-\sin z$ and $(\sin z{)}^{\mathrm{\prime }}=\cos z$ (Complex sine, cosine, hyperbolic sine, and hyperbolic cosine are entire with their standard derivatives).

Given: The annulus $A$, the function $f(z)=1\mathrm{/}z$, and the unit circle $\gamma$.

The modulus is multiplicative and satisfies the triangle inequality, hence $\mid \mathrm{\mid }z\mathrm{\mid }-\mathrm{\mid }w\mathrm{\mid }\mid \le \mathrm{\mid }z-w\mathrm{\mid }$ (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive).

A rational function is holomorphic wherever its denominator is nonzero (Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero).

Every nonzero complex number has a polar representation $r(\cos \theta +i\sin \theta )$ with $r>0$, and $\exp (i\theta )=\cos \theta +i\sin \theta$ for real $\theta$ (Every nonzero complex number has a unique polar form $r(\cos \theta +i\sin \theta )$ with $r>0$ and $-\pi <\theta \le \pi$, $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$).

A space is path-connected when each pair of points is joined by a continuous path, and every path-connected space is connected (Paths, path-connected spaces and path components, Every path-connected space is connected, and every path component lies inside a component).

A complex domain is a nonempty connected open subset of $\mathbb{C}$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$).

The integral of $z^{-1}$ around the positively oriented unit circle is $2\pi i$ (On a positively oriented circle about a, the integral of (z-a)^m is zero for every integer m except -1, and is 2 pi i for m=-1).

The complex exponential is entire and therefore continuous (The complex exponential is entire and its complex derivative is itself, Complex differentiability at a point implies continuity there).

Given: The punctured complex plane $U=\mathbb{C}\setminus \{0\}$ under the Euclidean identification of $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves.

For $n\ge 2$, the punctured Euclidean space ${\mathbb{R}}^n\setminus \{0\}$ is polygonally connected (For $n\ge 2$, the punctured space ${\mathbb{R}}^n\setminus \{0\}$ is polygonally connected).

Polygonal connectedness supplies paths, and every path-connected space is connected (Polygonal paths and polygonally connected subsets of ${\mathbb{R}}^n$, Paths, path-connected spaces and path components, Every path-connected space is connected, and every path component lies inside a component).

A complex domain is a nonempty connected open subset of $\mathbb{C}$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$).

A set is star-shaped with respect to $a$ only if the segment $(1-t)a+tz$ lies in the set for every member $z$ and every $0\le t\le 1$ (Star-shaped open subsets of Euclidean space).

The complex modulus satisfies $\mathrm{\mid }z+w\mathrm{\mid }\le \mathrm{\mid }z\mathrm{\mid }+\mathrm{\mid }w\mathrm{\mid }$ and vanishes exactly at zero (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive).

Given: The asserted need for a separate continuity hypothesis on the derivative.

Goursat's triangle theorem assumes only that $f$ is holomorphic on an open set containing the filled triangle and concludes that its boundary integral is zero; it explicitly makes no continuity assumption on $f^{\mathrm{\prime }}$ (Goursat's triangle theorem: a holomorphic function integrates to zero around every triangle contained in its domain).

Given: The whole complex plane $U=\mathbb{C}$, the function $f(z)=\bar{z}$, and the positively oriented unit circle $\gamma (t)=\exp (it)$.

Complex conjugation preserves differences and modulus, and $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$ (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive).

The whole Euclidean plane is convex, since every segment between two of its points remains in the plane (A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points).

The integral of $z^{-1}$ around the positively oriented unit circle is $2\pi i$ (On a positively oriented circle about a, the integral of (z-a)^m is zero for every integer m except -1, and is 2 pi i for m=-1).

A primitive $F$ of $f$ is holomorphic and satisfies $F^{\mathrm{\prime }}=f$ (A primitive of a complex function on an open set).

If $F$ is holomorphic, $F^{\mathrm{\prime }}$ is continuous, and $\gamma$ is closed and rectifiable, then ${\int }_{\gamma }{F}^{\mathrm{\prime }}=0$ (The integral of a continuous complex derivative over every closed rectifiable contour is zero).

The Euclidean plane ${\mathbb{R}}^2$ is connected (${\mathbb{R}}^n$ is polygonally connected, connected, locally path-connected and locally connected).

A complex domain is a nonempty connected open subset of $\mathbb{C}$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$).