= Complex analysis
{wiki}
Complex analysis studies <functions> of a <complex number>, especially <holomorphic functions>. Its distinctive feature is that a single complex <derivative> controls variation in every direction of the <complex plane>. The map $f(z)=z^2$ gives a first picture: straight coordinate lines become curved, yet small perpendicular arrows remain perpendicular wherever $f'(z)\ne0$.
\Image[/complex-square-map.png]
{title=The complex square bends a grid globally and acts near 1 + i as a rotation through 45 degrees and a scaling by twice the square root of two}
The bottom panels show displacements rather than absolute positions. Multiplication by $f'(1+i)=2+2i$ predicts the image of each small arrow; the exact image differs by $h^2$. The origin is exceptional because $f'(0)=0$. This distinction between local <conformal maps> and a <critical point> is central: <complex differentiability> imposes strong geometric structure, without making every <holomorphic function> globally one-to-one.
The remarkable next step is that <holomorphic functions> are determined inside a circle by their values on its boundary. Precisely, let $R>0$, let $D=\{z:|z-a|<R\}$, and suppose $f$ is a <holomorphic function> on an <open set> containing the closed disc $\overline D$. Orient its boundary counterclockwise. The <Cauchy integral formula> states that, for every $w\in D$,
$$
f(w)=\frac{1}{2\pi i}\oint_{|z-a|=R}\frac{f(z)}{z-w}\,dz.
$$
The <contour integral> combines boundary values with a kernel depending on the requested interior point. It is a reconstruction formula, not merely a trick for evaluating an <integral>. The hypothesis about the closed disc ensures that all boundary values and the <contour integral> are well defined. https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/b1a90df19f643b974555bbbb93138a48_MIT18_04S18_topic4.pdf[Orloff, MIT Topic 4, Theorem 4.1]{ref}
At the centre, substituting $z=a+Re^{i\theta}$ cancels the denominator against $dz=iRe^{i\theta}d\theta$:
$$
f(a)=\frac{1}{2\pi}\int_0^{2\pi}f(a+Re^{i\theta})\,d\theta.
$$
Thus the centre value is the angular average of the boundary values. For the <polynomial> $f(z)=z^2$, $a=1+i$ and $R=1/2$, expand the boundary data as
$$
f(a+Re^{i\theta})=2i+(1+i)e^{i\theta}+\tfrac14e^{2i\theta}.
$$
Both oscillating terms have zero average, leaving $f(a)=2i$. Away from the centre the weights change: on the <unit circle>,
$$
f(w)=\frac{1}{2\pi}\int_0^{2\pi}\frac{f(e^{i\theta})}{1-we^{-i\theta}}\,d\theta,\qquad |w|<1.
$$
With $f(z)=z^2$ and $w=1/2$, expanding the denominator as a <geometric series> leaves only its second-power term after integration, giving $f(1/2)=1/4$. This also explains how the <Cauchy integral formula> recovers an entire <Taylor series> from boundary data.
When a <holomorphic function> has <isolated singularities>, a different boundary calculation detects their <residues>. Suppose $\Gamma$ is a counterclockwise simple closed <contour> and $f$ is <holomorphic> on a neighbourhood of it and its interior except at finitely many interior <isolated singularities> $a_j$. The <residue theorem> gives
$$
\oint_\Gamma f(z)\,dz=2\pi i\sum_j\operatorname{Res}(f,a_j).
$$
A <residue> is the coefficient of $(z-a_j)^{-1}$ in the <Laurent series>. A short proof idea is to deform the <contour> to small circles around the <isolated singularities> using the <Cauchy integral theorem>. Integrating the <Laurent series> on each circle kills every power except $-1$. https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/3c58ce71ec6f7573804221d76158a0bd_MIT18_04S18_topic8.pdf[Orloff, MIT Topic 8, residue theorem]{ref}
Here is how a complex <pole> determines a real <improper integral>. Set $g(z)=1/(1+z^2)$ and use the <contour> consisting of the segment from $-R$ to $R$, followed by the upper semicircle $A_R$, with $R>1$. Only the <simple pole> $i$ lies inside, and
$$
\operatorname{Res}(g,i)=\lim_{z\to i}\frac{z-i}{(z-i)(z+i)}=\frac{1}{2i},\qquad \int_{-R}^R\frac{dx}{1+x^2}+\int_{A_R}g(z)\,dz=\pi.
$$
The arc must actually disappear. On $A_R$, the <reverse triangle inequality> gives $|1+z^2|\ge R^2-1$, and its length is $\pi R$, so
$$
\left|\int_{A_R}g(z)\,dz\right|\le\frac{\pi R}{R^2-1}\longrightarrow0.
$$
The real integrand is bounded by $x^{-2}$ for $|x|\ge1$, so both tails converge separately. Taking the limit therefore proves the ordinary <improper integral>
$$
\int_{-\infty}^{\infty}\frac{dx}{1+x^2}=\pi.
$$
For example, at $R=10$ the real segment contributes $2\arctan(10)\approx2.942255$, and the arc contributes $0.199337$, below the bound $10\pi/99\approx0.317333$. The usefulness of the <residue theorem> is that the same strategy works for many <rational functions> whose real <antiderivatives> are cumbersome: find the enclosed <poles>, compute their <residues>, and control the connecting arc. https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/f69bc227ae476839819fb18e5b283072_MIT18_04S18_topic9.pdf[Orloff, MIT Topic 9, contour decay estimates]{ref}
Historically, Cauchy's 1825 memoir on <integrals> with imaginary limits develops complex <contour integration> and its use in transforming and evaluating definite <integrals>. The opening sections define complex path <integrals>, discuss changing the path, and examine what happens near singular points. These are recognizable ancestors of the modern link between <holomorphic functions>, <contours> and <residues>. https://www.numdam.org/item/BSMA_1874__7__265_0.pdf[Cauchy, Mémoire sur les intégrales définies, prises entre des limites imaginaires, 1825; reprinted 1874, sections 1–6]{ref}
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