528 problems.
- \(g(\{|z|<r\})\subseteq f(\{|z|<r\})\) when \(f\) is injective holomorphic and \(f(0)=g(0)\)
- \(\sum_{k\in\mathbb{Z}}\frac{(-1)^k}{(k+a)^2}=\pi^2\cos(\pi a)\csc^2(\pi a)\) for \(a\in\mathbb{R}\setminus\mathbb{Z}\)
- An entire function with \(\min_{\abs z=r}\abs{f(z)}\) tending to a finite positive limit is constant
- \(\sum_{k\in\ZZ}(z-k)^{-2}=\pi^2/\sin^2(\pi z)\)
- \(\int_0^\infty x^{1/3}/(1+x^2)\,dx\)
- \(\int_\Gamma\Re(z)\,dz\) around the unit square
- \(\int_0^{2\pi}\frac{d\theta}{1+a^2-2a\cos\theta}\)
- The series of iterates \(\sum f^n\) of a non-rotation \(f\) analytic on \(\mathbb{D}\) with \(f(0)=0\) converges locally uniformly
- An entire function omitting a disk is constant
- Integrals over circles by Cauchy’s integral formula
- Mapping \(\DD^c \cap \HH\) conformally onto \(\HH\) with prescribed boundary values
- Once complex-differentiable functions are holomorphic
- Entire functions satisfying \(g(1-z)=1-g(z)\) are surjective
- Unique fractional linear transformation sending one circle to another with two prescribed values
- \(\abs{f(z)+f(-z)}\le2\abs z^2\) for a holomorphic self-map of \(\DD\) fixing \(0\)
- If \(f\) and \(\bar f\) are holomorphic on a domain, then \(f\) is constant
- \(\int_\RR\frac{\sin x}{x}\,dx\) by an indented semicircle and Jordan’s lemma
- Entire functions with \(\abs{f(z)}\ge e^{\abs z}\)
- A conformal equivalence from the quarter-disc to the first quadrant
- \(\int_{\partial\DD}\frac{e^z}{z^2}\,dz\)
- \(\sin(z)/z\) has a removable singularity at \(0\)
- A conformal map from \(\DD\) onto \(\HH\)
- An isolated singularity with \(\abs{f(z)}\le\abs z^{-1/2}\) is removable
- Entire doubly periodic functions are constant
- \(\sum_{k\geq 0}z^k/k!\) converges locally uniformly to \(e^z\)
- \(\sum_{k\geq 1}\frac{1}{k^2+a^2}=\frac{\pi\coth(\pi a)}{2a}-\frac{1}{2a^2}\) for \(a>0\)
- Locally uniform limits of holomorphic functions are holomorphic
- Radius of convergence of \(\sqrt{z}\) about \(4+3i\)
- \(\abs{f'(a)}\le1\) at a fixed point and \(\abs{f(0)}^2+\abs{f'(0)}^2\le1\) by Schwarz–Pick
- Area of \(f(D_r)\) is \(\pi\sum_{n=1}^\infty n|c_n|^2 r^{2n}\) for univalent \(f(z)=\sum c_n z^n\)
- Uniform convergence of \(\sin(nx)/(1+nx)\)
- An entire function with a vanishing Taylor coefficient at every point is a polynomial
- Nonvanishing holomorphic functions of modulus one on the circle are constant
- \(\abs{f''(0)}\le2\) for a self-map of \(\DD\) with \(f(0)=f'(0)=0\)
- A conformal map from the upper half-disk onto \(\HH\)
- Real-valued holomorphic functions are constant
- No continuous square root on \(\CC\)
- Harmonic functions on \(\DD\) with \(u(1/2)=2\) and \(\abs u\ge2\)
- Uniform limits preserve continuity and uniform continuity; pointwise limits need not
- \(\sum_{k\in\mathbb{Z}}\frac{1}{(k-1/2)^2}=\pi^2\)
- Principal parts of \(P/Q\) at simple and double poles
- No conformal bijection from the punctured disk onto the annulus \(1<\abs z<2\)
- Complex line integrals and the ML bound for \(\log(z)/z^2\) on \(\abs z=R\)
- Taylor expansion of \(z^2\cos(z/3)\)
- \(\int_{\mathbb R} e^{ax}\operatorname{sech}(x)\,dx\)
- \(\int_\RR\frac{dx}{(1+x^2)^{n+1}}\)
- \(\int_\RR\frac{dx}{(1+x^2)^2}\)
- A pole of order \(n\) of \(f\) is a pole of order \(n+k\) of \(f^{(k)}\)
- Residues of \(z^3/(z^2+1)\) from its principal parts
- \(\sum_{k\in\ZZ}\frac{1}{k^2+a^2}=\frac{\pi\coth(\pi a)}{a}\) for \(a>0\)
- \(\int_\RR\frac{x\sin x}{1+x^2}\,dx\) by Jordan’s lemma
- \(\int_\RR\frac{\cos x}{x+i}\,dx\) by Jordan’s lemma
- An entire function with \(f(z)/z\to0\) is constant
- A harmonic conjugate of \(x^3-3xy^2-x-y\)
- \(\int_{-\pi}^{\pi}\frac{d\theta}{1+\sin^2\theta}\)
- \(u^2-v^2\), \(uv\) and \(u_x\) are harmonic for harmonic conjugates \(u,v\)
- A \(C^1\) function satisfies the Cauchy–Riemann equations if and only if it is holomorphic
- Entire functions with a pole at infinity are polynomials
- Entire \(O(|z|^p)\) functions are polynomials of degree at most \(\lfloor p\rfloor\)
- Power series of holomorphic functions on the disk converge uniformly on compacta
- Laurent series of \(\frac{1}{z(z-1)}\) on \(\abs z<1\) and on \(\abs z>1\)
- Polynomials cannot uniformly approximate \(z^{-m}\) on an annulus
- A nonzero analytic function on the disk with infinitely many zeros
- Entire functions with \(\abs f\le\abs g\) satisfy \(f=cg\)
- Uniformly bounded derivatives imply equicontinuity
- Automorphisms of \(\CP^1\)
- Differentiable contractions are uniformly continuous
- Meromorphic functions on \(\mathbb{CP}^1\)
- An entire function with \(\abs{f(z)}\le M\abs z^n\) for large \(\abs z\) is a polynomial of degree at most \(n\)
- \(\|f\|_{(\infty,s)}\le c\|f\|_{(1,r)}\) for holomorphic \(f\)
- Hadamard expansion of \(\sin\)
- \(\int_\RR\frac{xe^{2ix}}{x^2-1}\,dx\)
- Laurent series of \(\frac{1}{(z-3)(z-1)}\) on the three annuli about \(0\)
- Schwarz–Pick lemma
- Heine–Cantor theorem: a continuous map on a compact metric space is uniformly continuous
- Pointwise limits of entire functions, uniform on segments, are entire and compactly convergent
- \(\int_0^1\frac{dx}{\sqrt{x^2-1}}\) by a dogbone contour
- Isolated zeros of \(f'\), and \(f'=g'\) implies \(f-g\) is constant
- A function with bounded derivative is uniformly continuous
- Power series are continuous
- Laurent series of \(e^{1/z}\cos(1/z)\) at its essential singularity \(0\)
- \(\int_0^\infty\frac{\log x}{(1+x^2)^2}\,dx\) by a keyhole contour
- \(f(z)=-\frac12\bigl(z+\frac1z\bigr)\) maps the upper half-disk conformally onto the upper half-plane
- \(\Res_{z=0}\frac{1}{z^2\sin z}\)
- \(\abs{f(z)}\le\abs{\frac{z-i}{z+i}}\) for holomorphic \(f:\HH\to\DD\) with \(f(i)=0\)
- \(\int_0^\infty\frac{\log x}{1+x^2}\,dx\)
- Convergence of \(\frac1z\sum_{k=1}^\infty\frac{z^k}{k}\) on \(S^1\setminus\{1\}\) by summation by parts
- A zero of multiplicity \(n\) is a point where \(f,\ldots,f^{(n-1)}\) vanish and \(f^{(n)}\) does not
- Analyticity of \(\Gamma\) on \(\Re(s)>0\) and Euler’s reflection formula
- \(\int_0^{\infty}\frac{\cos x}{x^2+b^2}\,dx=\frac{\pi e^{-b}}{2b}\)
- Radii of convergence of the principal \(\sqrt z\) about \(4+3i\) and \(-4+3i\)
- A self-map of \(\DD\) with \(f(1/2)=3/4\) and \(f'(1/2)=2/3\)
- Sharp bound on \(\abs{f'(0)}\) for \(f:\DD\to\theset{\Re w>0}\) with \(f(0)=2\)
- Holomorphic functions on a domain: \(fg\equiv 0\) implies \(f\equiv 0\) or \(g\equiv 0\)
- Holomorphic functions of unit modulus on the circle are constant
- Orders of the zeros of \((e^z-1)^3\)
- A holomorphic function with a vanishing Taylor coefficient at every point is a polynomial
- Mapping \(\DD^c \intersect \HH\) to \(\HH\) via cross-ratios
- Schwarz–Pick lemma
- The family \(\{z^k\}\) on \([0,1]\) is not equicontinuous
- Laurent series and residue of \(\cos(1-1/z)\) at \(0\)
- \(\int_{-1}^1 ((x-a)\sqrt{1-x^2})^{-1}\,dx\)
- Bound on \(\abs{f(2i)}\) for \(f:\HH\to\DD\) with \(f(i)=0\)
- There is no entire function with \(|f(z)|\ge|z|+1\) for all \(z\)
- Sums and products of poles and essential singularities
- A conformal map from the lens \(\{|z-\lambda|<1\}\cap\{|z-\bar\lambda|<1\}\) onto \(\mathbb{D}\), for \(\lambda=\frac12(1+i\sqrt{3})\)
- A fractional linear map sending \(i\RR\) onto \(\abs{z-\frac12}=\frac12\)
- An entire function whose image misses a disk is constant
- The integral of \(\log x/(1+x^a)\) over \([0,\infty)\)
- Conformal maps of lunes, slit disks, and half-disks
- An injective holomorphic function has nonvanishing derivative
- \(B(z,w)=\frac{\Gamma(z)\Gamma(w)}{\Gamma(z+w)}\)
- A bounded harmonic function on \(\RR^2\) is constant
- Entire functions with \(|f(z)/z^{n}|\) bounded at infinity are polynomials of degree at most \(n\)
- \(\int_\RR\frac{\cos x}{x^2+1}\,dx\) by a semicircular contour
- A sequence with uniformly bounded derivatives on \([0,1]\) is equicontinuous
- The residue of \(1/(z^n+1)\) at a root of unity
- A self-map of \(\DD\) is bounded by the Blaschke product of its zeros
- A holomorphic function with a vanishing Taylor coefficient at every point is a polynomial
- \(\int_\RR\frac{dx}{x^4+1}\) by a semicircle
- Constancy on a closed subdisk of \(\mathbb{D}\) implies constancy on \(\mathbb{D}\)
- The integral of \(1/(x^4+1)\) over \(\mathbb{R}\)
- \(\int_0^{2\pi}\frac{d\theta}{a+b\cos\theta}\)
- Extended Liouville theorem
- Orders of the poles of \(\frac{1}{z\sin z}\) and \(\frac{e^{z^2}-1}{z^4}\) at \(0\)
- Mobius transformations as cross-ratios
- Bound on \(\abs{f(\frac{1+i}{2})}\) for a self-map of \(\DD\) with \(\abs{f}\le\abs{e^z}\) on \(\abs z=1\)
- A singularity is removable for \(f\) if and only if it is removable for \(f'\)
- \(\abs{e^z}=e^{\Re z}\)
- The Dirichlet function is nowhere continuous and not Riemann integrable
- \(x/(x^2+4x+13)^2\)
- \(\log(x) / x^2+a^2\)
- Injective conformal maps from \(\{\Re z>0,\ |z-1|>1\}\) onto \(\mathbb{D}\)
- Bounded derivatives imply removable singularities
- Injective holomorphic maps have nonvanishing derivative
- \(1/x^2+3x+2\)
- Compactly convergent series of holomorphic functions are holomorphic
- \(\sqrt{x^2-1}\)
- Finitely many poles
- \(\frac{|f(0)|-|z|}{1+|f(0)||z|}\le|f(z)|\le\frac{|f(0)|+|z|}{1-|f(0)||z|}\) for holomorphic \(f:\mathbb{D}\to\mathbb{D}\)
- Schwarz reflection principle on a half-disk
- Constructing functions with specified singularities
- A non-equicontinuous sequence
- Essential singularities
- Primitive in the complement of a branch cut
- Cauchy formula and principal parts
- Estimating and conformal maps
- Schwarz with domain/codomain scaled
- \(f(z)= u(x,y) +i v(x,y)\) be analytic in a domain
- Hadamard’s ill-posed Cauchy problem for Laplace’s equation
- Constant boundary modulus implies constancy or an interior zero
- Cauchy–Riemann equations and elementary complex analysis
- The false uniform-continuity claim for \(x^{-n}\) near \(0\)
- Primitives imply vanishing integral
- Locally uniform limit theorem for holomorphic functions
- Singularities of \(1\over e^z - 1\)
- Sum formulas: \(1/n^2\)
- An entire function \(h\) with \(\frac{\pi^2}{\sin^2(\pi z)}=\sum_{k\in\mathbb{Z}}\frac{1}{(z-k)^2}+h(z)\)
- Complement of a segment
- Entire functions with \(|f(z)|\le|\sin z|\) are \(c\sin z\) with \(|c|\le 1\)
- A bijective analytic map on \(\mathbb{C}\setminus\{z_0\}\), bounded off a neighborhood of \(z_0\), is a Möbius transformation \(\frac{az+b}{cz+d}\) with \(c\neq 0\)
- Rudin 10.4
- Convergence of \(\prod_{n\in\mathbb{Z}}(1+a_n)\) when \(\{a_n\}\in\ell_1(\mathbb{Z})\)
- \(\frac{\pi^2}{\sin^2\pi z}=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n)^2}\) by matching principal parts and boundedness of the difference
- Uniform continuity of \(x^n\)
- Holomorphic iff delbar vanishes
- Trig identities
- Polar CR equations
- Polynomial growth of entire functions and Liouville for bounded real part
- \(\frac{\pi^2}{\sin^2(\pi z)}=\sum_{n\in\mathbb{Z}}\frac{1}{(z-n)^2}\) by matching singularities, principal parts, and decay in a period strip
- Half disc to full disc
- Cancelling poles
- Entire functions with \(f(z)/z^{n}\to 0\) at infinity are polynomials of degree at most \(n-1\)
- Sinc Function
- Complement of the disc to \(\mathbb{H}\)
- The logarithmic derivative of a product: \(\frac{(fg)'}{fg}=\frac{f'}{f}+\frac{g'}{g}\)
- Standard sector
- Holomorphic functions with vanishing derivative are constant on connected domains
- Laurent expanding tricky exponentials
- Residues at infinity
- Special case of bijections
- Uniform limits of derivatives, term-by-term differentiation
- Bounded Complex Analytic Functions form a Banach Space
- Cauchy integral formula for coefficients
- Entire functions with poles at infinity
- Cauchy formula and \(\sinh\)
- Holomorphic functions satisfying \(f^{2}=\overline{f}\) are constant
- Integral computation
- Integrating \(z^k\) around \(S^1\) is the source of residue theory
- If poles accumulate at \(a\), then every \(w\in\mathbb{C}\) is a sequential limit of \(f\), versus Casorati–Weierstrass
- Power series converge uniformly on their radius of convergence
- Limit point of zeros is an essential singularity
- The integral of \(1/(1+x^2)^2\) over \(\mathbb{R}\)
- \(\cos(x) / x^2 + b^2\)
- Expansion for a reciprocal
- Entire functions with \(f(z)\to\infty\) as \(z\to\infty\) are polynomials
- \(\bigl|\frac{w-z}{1-\bar{w}z}\bigr|\) on the closed disk, and \(z\mapsto\frac{w-z}{1-\bar{w}z}\) is an automorphism of \(\mathbb{D}\)
- \(z^5+3z+1\) has five zeros in \(|z|\leq 2\)
- \(\|f\|_{(\infty,s)}\le c\|f\|_{(1,r)}\) for holomorphic \(f\) near \(D_r(z_0)\)
- Fixed point convergence
- Area of the image of an annulus under a univalent Laurent series
- A real-valued holomorphic function is constant
- Holomorphic automorphisms of \(\CC\) are the maps \(z\mapsto az+b\) with \(a\neq 0\)
- Images of sequences of poles converge to anything
- Residue from Laurent expansion: \(1/(z - \sin(z))\)
- Entire functions with a vanishing Taylor coefficient at every point are polynomials
- Zero derivative implies constant
- An entire function with bounded \(n\)th derivative is a polynomial of degree at most \(n\)
- Holomorphic functions have harmonic components
- Residues at \(\infty\)
- Entire functions with \(|f(z)|>1\) are constant
- A holomorphic function on \(\Delta^*\) with nonnegative real part has a removable singularity at \(0\)
- \(1/x^4+1\), half-line
- Equicontinuity + pointwise convergence implies uniform convergence
- Product of sines \(\sin(k\pi/n)\) via roots of unity
- Lune between circles
- Injectivity relates to derivatives
- Entire functions for which \(f(1/z)\) has a pole at \(0\) are polynomials
- Simple zeros of \(\sin(\pi z)\) at the integers, and residues of \(1/\sin(\pi z)\)
- Computing \(\zeta(2)\) by integration
- Laurent expansions about different points
- Equality of different integrals
- The logarithmic derivative of a product: \(\frac{(fg)'}{fg}=\frac{f'}{f}+\frac{g'}{g}\)
- Lune with only one intersection point
- Convergence of a \(\ZZ\dash\)index series
- Fixed points: multiple
- Removable singularity at infinity iff constant
- Holomorphic functions with vanishing derivative are constant
- A conformal map from \(\HH\) onto \(\{-\pi/2<\Re w<\pi/2,\,\Im w>0\}\)
- Disc to upper half-plane, cross-ratio
- Zeros of \(\sin(\pi z)\) and singularities of \(\csc(\pi z)\)
- Residues and classifying singularities
- The locus \(\Arg\qty{z-a\over z-b}=\theta\): circular arcs through \(a\) and \(b\)
- Bounded holomorphic functions form a Banach space
- Entire functions agreeing with \(e^x\) on \(\RR\)
- Working with conjugates
- An entire function with values in \(\HH\) is constant
- Holomorphic functions have discrete zeros
- Entire functions with \(\lim_{z\to\infty}f(z)=\infty\) are polynomials
- Detecting injectivity using derivatives
- Poles of elliptic functions
- Lune, one intersection
- Finding complex roots
- Schwarz reflection and holomorphic functions real on the unit circle
- Application of summation by parts
- Combining with Rouche
- Laurent expanding exponentials
- Classifying singularities
- Automorphisms of \(\CC\)
- \(z+3+2e^z\) has one root in \(\{\Re z\le 0\}\)
- Showing singularities are removable
- Liouville
- Uniform convergence \(f_n\to f\) implies \(\int_\gamma f_n\to\int_\gamma f\)
- Uniform limit theorem for holomorphic functions
- \(1/x\sqrt{x^2-1}\)
- \(\sin z=z\) has infinitely many solutions in \(\CC\)
- Uniform limit of holomorphic derivatives is the derivative
- \(x^\alpha/(x+1)^2\)
- Disc minus a slit
- A conformal map from the upper half-disc to the upper half-plane
- An analytic function sending a simple closed curve into \(\RR\) is constant
- Analytic self-maps of the disc with a zero of order \(k\) at \(0\) and \(|f|\to 1\) at the boundary
- Uniformly bounded but not equicontinuous
- Residue of \(1/z^2 + 1\)
- \(e^{x/2} / 1+e^x,\) replication
- Residue at a pole of order \(m\), and \(\oint_C e^\tau/(\tau^2+\pi^2)^2\,d\tau\)
- Conformal maps to arbitrary points
- \(1/1+a^2+2a\cos(\theta)\), Poisson kernels
- The integral of \(1/(x\sqrt{x^2-1})\) over \([1,\infty)\)
- A holomorphic function attaining a minimum of \(|f|\) vanishes there or is constant
- Purely imaginary if on circle
- Entire functions with nonnegative real part are constant
- More juggling
- \(\sin(x)/x\) and \(\cos(x)/x\)
- Radius of convergence
- Forcing a map to be the identity
- Primitives imply vanishing integral
- Series convergence on the circle by summation by parts
- \(\int_{-\infty}^\infty\frac{1+x^2}{1+x^4}\,dx\)
- An entire function with \(|f(z)|\le|z|^{1/2}\) for \(|z|>10\) is constant
- Laurent expansions of \(\frac{z+1}{z(z-1)^2}\) about \(z=0\) and \(z=1\)
- \(e^{-\pi\xi^2}=\int_{-\infty}^\infty e^{-\pi x^2}e^{2\pi i x\xi}\,dx\) for all \(\xi\in\CC\)
- Liouville’s theorem
- Unique zero of \(z-a-qf(z)\), residue formula, and series for \(1/F\)
- Integrals over receding vertical segments when \(f(x+iy)\) tends uniformly to a constant
- Possible holomorphic interpolations of sequences accumulating at \(0\)
- Reverse triangle inequality \(|z+w|\ge\bigl||z|-|w|\bigr|\)
- \(\int_0^\infty\frac{x^{a-1}}{1+x^n}\,dx=\frac{\pi}{n\sin(\pi a/n)}\) and \(\int_0^\infty\frac{\log x}{(1+x^2)^2}\,dx=-\frac{\pi}{4}\)
- Dirichlet, Mellin, and logarithmic integrals by residues
- \(c_n\neq 0\) for large \(n\) and \(c_n/c_{n+1}\to z_0\) for a pole on the unit circle
- Implicit function theorem for \(f(s,t)=ps^3-6st+t^2\) at \((1,3)\), over \(\RR\) and \(\CC\)
- Schwarz reflection across the real axis, and holomorphic functions real on the circle
- Roots of \(z^3+2z+4\) lie outside the unit circle
- Existence of multiple values in branches
- A conformal map from the upper half-plane onto \(\{-\pi/2<\Re w<\pi/2,\,\Im w>0\}\)
- Polynomials do not converge uniformly to \(1/z\) on \(S^1\)
- Unique fixed points
- The fundamental theorem of algebra via the maximum modulus principle
- Zeros of a polynomial in the annulus one to two
- Fixed points of holomorphic self-maps of the disk
- Fundamental Theorem of Algebra via Rouché’s theorem
- Entire functions with \(|f(z)|\le|z|^{1/2}\) for \(|z|>10\) are constant
- An entire function with a vanishing Taylor coefficient at every point is a polynomial
- Laurent expansions of \(\frac{z+1}{z(z-1)}\) about \(0\) and \(1\)
- Unique solution of \(\alpha ze^z=1\) in the unit disc when \(|\alpha|>e\)
- Analyticity of \(z\mapsto\overline{f(\overline{z})}\)
- A sharp bound on \(|f'(0)|\) for maps to the right half-plane with \(f(0)=2\)
- A conformal map from \(\{|z|<1,\ |z-1/2|>1/2\}\) onto \(\DD\)
- Monotone pointwise convergence to \(0\) of continuous functions on \([0,1]\) is uniform
- \(\int_T f=0\) if \(f\) is holomorphic and bounded near a puncture inside the triangle \(T\)
- Laurent series of \(\frac{1}{(z-1)(z-2)}\) in the annulus \(1<|z|<2\)
- Conformal map from \(\{|z|<1,\ |z-1/2|>1/2\}\) onto the unit disc
- A holomorphic self-map of a bounded open set with a fixed point of derivative \(1\) is linear
- Number of solutions of \(e^z=az^n\) in the unit disc when \(|a|>e\) and when \(|a|<1/e\)
- Laurent series of \(\frac{1}{z}+\frac{1}{z^2-1}\) on the largest annuli of validity
- \(\int_0^{2\pi}\frac{1}{a+\cos\theta}\,d\theta=\frac{2\pi}{\sqrt{a^2-1}}\) for \(a>1\)
- Power series of radius \(1\): convergence on the circle versus analyticity on the circle
- Schwarz lemma and \(g(\{|z|<r\})\subseteq f(\{|z|<r\})\) for holomorphic \(f,g:\DD\to\Omega\) with \(f\) injective and \(f(0)=g(0)\)
- Unique fractional linear transformation sending a circle \(C\) to \(C'\) with \(f(z_1)=z'_1\) and \(f(z_2)=z'_2\)
- Laurent expansions of a rational function and their domains
- The parallelogram identity for complex numbers and its geometric meaning
- An entire function with \(|f(z)|\le A|z|^N\) at infinity is a polynomial of degree at most \(N\)
- Power series are continuous on their disk of convergence
- Constancy of holomorphic functions of constant modulus, real part, argument, or conjugate
- Rouché’s theorem and \(\max_{|z|=1}|P(z)|\ge 1\) for a monic polynomial \(P\)
- A power series convergent at \(z_0\neq 0\) converges absolutely for \(|z|<|z_0|\) and uniformly on \(|z|\le r<|z_0|\)
- A conformal map from \(\CC\setminus[1,\infty)\) onto \(\DD\)
- The only entire function with \(|f(z)|\le|z|^{1/2}\) for \(|z|>10\) is \(f\equiv 0\)
- Self-maps of \(\DD\) unimodular on the circle are finite Blaschke products
- Entire functions of quadratic growth are polynomials of degree at most \(2\)
- Uniform convergence of \(\sum|a_n'|\) on compacta from that of \(\sum|a_n|\)
- If \(fg=0\) on a domain then \(f\equiv 0\) or \(g\equiv 0\)
- A holomorphic function real-valued on a closed curve is constant
- A function holomorphic on \(0<|z|<1\) with \(\int_{|z|=r}f=0\) for all \(r<1\), but not holomorphic at \(0\)
- \(2|f'(0)|\le\operatorname{diam} f(\DD)\), with equality iff \(f\) is linear
- A function continuous on \(\mathbb{C}\) and holomorphic off \(\mathbb{R}\) is entire
- Analytic functions of equal modulus differ by a unimodular constant
- Roots of \(z^3+2z+4\) lie outside the unit disk
- Laurent expansions of \(e^{1/z}\) and \(\cos(1/z)\) about \(0\)
- Cauchy’s formula for an exterior region
- Distance between disjoint compact sets in \(\RR^n\) is attained
- Principal part of \(P/Q\) at a simple or double root of \(Q\)
- \(\int_0^\infty\frac{x^{a-1}}{1+x^n}\,dx=\frac{\pi}{n\sin(a\pi/n)}\) for \(0<a\le n\)
- Holomorphic self-maps of the disk with a zero of order \(k\) at \(0\) and \(|f|\to 1\) at the boundary
- Holomorphic functions on the disk expand as power series uniformly on compact subsets
- \(|z|^2\) is complex-differentiable only at \(0\)
- Conformal map from a slit lens to the upper half plane
- Number of solutions of \(e^z=az^n\) in the unit disk
- (a) \(f(z)= u(x,y) +i v(x,y)\) be analytic in a domain
- \(\int_0^\infty\frac{dx}{(1+x^2)(1+9x^2)}\)
- The cubic \(z^3+3z^2+bz+b^2\) has exactly two roots in the unit disk when \(|b|<1\)
- \(\int_\gamma\frac{dz}{(z-\alpha)(z-\beta)}=\frac{2\pi i}{\alpha-\beta}\) without Cauchy’s formula
- \(\sin z/z\) has no poles
- A conformal map from \(\DD\setminus[0,1)\) to \(\DD\)
- Implicit function theorem from the inverse function theorem
- The union of intersecting connected sets is connected
- Schwarz reflection of a holomorphic function across the diameter of the disk
- An injective conformal map from \(\{|z|<1\}\setminus\{|z-1/4|\le 1/4\}\) onto an annulus
- Area of \(f(\{|z|<r\})\) is \(\pi\sum_{n=1}^\infty n|c_n|^2 r^{2n}\) for univalent \(f(z)=\sum_{n=0}^\infty c_n z^n\)
- Area of \(f(\{r<|z|<R\})\) is \(\pi\sum_{n=-\infty}^\infty n|c_n|^2(R^{2n}-r^{2n})\) for a univalent Laurent series
- Liouville’s theorem via \(\lim_{R\to\infty}\int_{|z|=R}\frac{f(z)}{(z-a)(z-b)}\,dz\)
- \(\frac1z\sum_{k=1}^\infty z^k/k\) converges on \(S^1\setminus\{1\}\)
- Uniqueness of Laurent series coefficients
- A uniformly continuous map preserves uniform convergence under composition
- The distance between disjoint compact sets in \(\mathbb{R}^n\) is attained
- Convergence of \(\sum nz^n\), \(\sum z^n/n^2\), and \(\sum z^n/n\) on the unit circle
- Explicit Rouché, \(\mathbb{D}\)
- A holomorphic function on \(\DD\setminus\{0\}\) with \(|f(z)|\le\log(1/|z|)\) is identically zero
- Liouville’s theorem via \(\lim_{R\to\infty}\int_{|z|=R}\frac{f(z)}{(z-a)(z-b)}\,dz\) for bounded entire \(f\)
- Riemann integrability of \(g\) when \(|g(x)-g(y)|\le|f(x)-f(y)|\) for integrable \(f\)
- The disk automorphism \(z\mapsto\frac{w-z}{1-\bar{w}z}\)
- A power series may be re-expanded about any point of its disk of convergence
- A conformal map from \(\DD\) to \(\HH\)
- Uniform approximation of holomorphic functions by polynomials
- The truncated exponential \(\sum_{k=0}^n z^k/k!\) has no zeros in the open unit disk
- Weierstrass theorem on locally uniform limits of holomorphic functions
- Holomorphy of \(\Gamma\) on \(\operatorname{Re}s>0\) and the reflection formula \(\Gamma(s)\Gamma(1-s)=\pi/\sin(\pi s)\)
- Holomorphic interpolation at sequences accumulating at zero
- Uniformly continuous functions preserve uniform convergence of sequences
- The maps \(z\mapsto(w-z)/(1-\bar{w}z)\) send the disc to itself
- Explicit Rouché, half-plane
- Integrals of \(z^n\) and \(1/((z-a)(z-b))\) on circles, without Cauchy’s theorem
- Cauchy integral formula for a function holomorphic outside \(\gamma\) with limit \(A\) at infinity
- Images of circles \(|z|=r\) and rays \(\arg z=\theta_0\) under \(z+1/z\)
- \(\int_0^\infty\frac{x^2}{(x^2+a^2)^2}\,dx\) for \(a>0\)
- Finite Blaschke products of degree \(n\) take every value \(n\) times in the disc
- Positive integers \(n\) with \((1+i)^n=(1-i)^n\)
- Cauchy’s theorem via Green’s theorem, and Goursat’s theorem
- Uniform convergence of \(\sum\sin(nz)/2^n\) on \(\{\Im z<\ln 2\}\)
- Schwarz–Pick lemma
- \(\int_0^{2\pi}\frac{d\theta}{(a+b\cos\theta)^2}\) for \(a>b>0\)
- \(\int_0^\infty\frac{x^{a-1}}{1+x^n}\,dx\) for \(0<a<n\) and \(\int_0^\infty\frac{\log x}{(1+x^2)^2}\,dx\)
- Number of roots of \(z^7-4z^3-1\) in \(|z|<1\)
- A fractional linear transformation \(\HH\to\DD\) and the image of the first quadrant
- A conformal map from \(\CC\setminus(-\infty,0]\) onto \(\DD\)
- Invert \(2z-1\), one coefficient at a time
- Polynomial approximation on \(\bar\DD\) and entire functions with a vanishing Taylor coefficient
- \(\int_0^\infty\frac{x\sin x}{x^2+a^2}\,dx\)
- \(z\sin z=a\) has only real solutions
- \(\int_C \frac{dz}{f(z)-f(z_0)}=\frac{2\pi i}{f'(z_0)}\) for a small circle about a noncritical point
- Riemann integrability under \(|g(x)-g(y)|\le|f(x)-f(y)|\)
- \(\frac{e^{ny}-e^{-ny}}{2n^2}\sin nx\) solves the Laplace Cauchy problem on the disk with data \(\sin(nx)/n\), and \(\limsup_n|u|=\infty\) at some points
- An entire function with \(f(z)/z^n\to 0\) at infinity is a polynomial of degree at most \(n-1\)
- Zeros of the truncated exponential \(P_n\) and of \(P_n-1\) in \(|z|<10\)
- Once complex differentiable implies holomorphic
- \(\lim_{r\to 0}\int_{\gamma_r}f=iA\beta_0\) when \((z-a)f(z)\to A\)
- Dini’s theorem: monotone continuous functions decreasing to \(0\) on \([0,1]\) converge uniformly
- Laurent series and singularities at \(0\) of \(\sin^2(z)/z\), \(z\exp(1/z^2)\), and \(1/(z(4-z))\)
- Identity theorem and Schwarz reflection for harmonic functions
- Conformal maps of lunes, half-disks, and slit disks
- Characterizations of Möbius transformations preserving \(\RR\cup\{\infty\}\)
- A conformal map from \(\{\Re z>0,\ |z-i|>1\}\) onto \(\HH\)
- The Fourier transform of a compactly supported continuous function is entire
- \(\int_{\RR}\frac{dx}{1+x^4}\) and the poles of \(\frac{1}{1+z^4}\)
- \(z^2\) is uniformly continuous on \(|z|<R\) but not on \(\CC\)
- Midpoint recurrence \(x_n=\frac{x_{n-1}+x_{n-2}}{2}\) converges to \(\frac{a+2b}{3}\)
- Green’s theorem for rectangles and Cauchy’s theorem on a rectangle
- Tangent from polynomial long division
- Continuous on \(\CC\) and holomorphic off \(\RR\) implies entire
- \(\int_\Delta f=0\) for triangles when \(f\) is holomorphic off one point and bounded nearby
- Log properties can fail
- Integral of \(f\) over a receding vertical segment equals \(iAb\)
- Cayley map \(\mathbb{H}\to\mathbb{D}\) and the image of the first quadrant
- Uncountable \(E\subset[0,1]\) is uncountable on both sides of some \(t\)
- Blaschke factors as automorphisms of the disk
- \(az^n+z+1\) has a root in \(|z|\le 2\)
- \(\int_0^\infty\frac{\sqrt{x}}{1+x^2}\,dx\)
- Polynomials cannot converge uniformly to \(1/z\) on \(S^1\)
- The union of two intersecting connected sets is connected
- Maximum modulus principle on the interior of a simple closed contour
- A holomorphic function is locally \(m\)-to-\(1\) near a zero of multiplicity \(m\)
- Local mapping theorem: a zero of multiplicity \(m\) splits into \(m\) distinct nearby preimages
- Number of zeros of \(z^3-z+1\) in the right half-plane
- An everywhere differentiable function whose derivative is discontinuous at \(0\)
- Power series of \(\tan z\) via geometric-series inversion
- Compactly uniform limits of holomorphic functions are holomorphic
- The ring of analytic functions on a domain is an integral domain
- Cauchy-Riemann equations in polar coordinates
- Triangle integrals vanish if \(f\) is bounded near an isolated singularity
- Cauchy property and limit of the midpoint recurrence \(x_n=(x_{n-1}+x_{n-2})/2\)
- \(\int_0^\infty\frac{\cos x}{(x^2+a^2)^2}\,dx\) for \(a>0\)
- Entire functions with a pole at \(\infty\) are the nonconstant polynomials
- Invert \(\cos(z)\), one coefficient at a tiem
- Uniform convergence of \(\sum|a_n|\) on compacta implies that of \(\sum|a_n'|\)
- \(\int_0^\infty\frac{\cos x}{x^2+b^2}\,dx=\frac{\pi e^{-b}}{2b}\)
- Analyticity of \(\int_\gamma\frac{g(w)}{w-z}\,dw\) off a smooth curve \(\gamma\)
- Holomorphic functions of modulus one on the circle are finite Blaschke products
- An entire function with \(f(z)/z\to 0\) as \(z\to\infty\) is constant
- A continuous function vanishing at \(\pm\infty\) is uniformly continuous
- One root of \(z^4+2z^3-2z+10\) in each open quadrant
- \(\int_T f=0\) when \(f\) is holomorphic and bounded near an interior puncture of \(T\)
- A conformal map from the strip \(\{0<\Im z<1\}\) onto \(\HH\)
- A conformal map from the lens \(|z-1|<2\cap|z+1|<2\) onto \(\HH\)
- Complex numbers on one side of a line through the origin have nonzero sum and sum of reciprocals
- \(f=u+iv\) is complex differentiable at \(z_0\) iff \(\lim_{r\to 0}\frac{1}{\pi r^2}\int_{|z-z_0|=r}f(z)\,dz=0\)
- \(\sum_{k=1}^{n-1}\cos\frac{2\pi k}{n}=-1\) and \(\sum_{k=1}^{n-1}\sin\frac{2\pi k}{n}=0\), via roots of unity
- Truncated exponentials \(\sum_{k=0}^n z^k/k!\) have no zeros in the unit disk
- Reverse triangle inequality \(|z-w|\ge\bigl||z|-|w|\bigr|\) and equality cases
- Continuity, differentiability, and \(C^1\) regularity of \(\frac{xy}{\sqrt{x^2+y^2}}\)
- Weierstrass’s theorem: locally uniform limits of holomorphic functions are holomorphic
- \(\int_{-\infty}^{\infty}\frac{e^{-2\pi ix\xi}}{\cosh\pi x}\,dx\)
- Horizontal strip to \(\mathbb{H}\)
- Holomorphic functions in a neighborhood of \(D_r(z_0)\)
- Entire functions of polynomial growth, sector vanishing on the disc, products of distances on \(S^1\), and constancy when the real part is bounded
- Uniform compact convergence of \(\sum f^{n}\) for analytic \(f\) on \(\DD\) with \(f(0)=0\) not a rotation
- If \(\lim_{n\to\infty}|a_{n+1}|/|a_n|=L\) then \(\lim_{n\to\infty}|a_n|^{1/n}=L\)
- An analytic function is constant if \(|f|\), \(\Re(f)\), or \(\arg(f)\) is constant, or if \(\overline{f}\) is analytic
- An everywhere differentiable function with \(f'\) discontinuous at \(0\)
- \(L^\infty\) and \(L^1\) disc norms of holomorphic functions
- \(z^3+3z^2+bz+b^2\) has exactly two roots in the unit disk when \(|b|<1\)
- Cauchy-Riemann equations in polar coordinates and the holomorphicity and discontinuity of \(\Log z\)
- \(\pi^2/\sin^2(\pi z)=\sum_{n\in\mathbb{Z}}(z-n)^{-2}\)
- An analytic self-map of the disk with \(|f|=1\) on the circle is a finite Blaschke product
- \(\bigl(\frac12(-1+\sqrt{3}i)\bigr)^n+\bigl(\frac12(-1-\sqrt{3}i)\bigr)^n\) equals \(2\) if \(3\mid n\) and \(-1\) otherwise
- Analyticity of \(\int_\gamma \frac{g(w)}{w-z}\,dw\) on \(\CC\setminus\gamma\)
- \(\int_0^\infty\frac{dx}{(1+x^n)^2}\), \(\int_0^\infty\frac{\cos x}{(x^2+a^2)^2}\,dx\), \(\int_0^\pi\frac{d\theta}{a+\sin\theta}\), \(\int_0^{\pi/2}\frac{d\theta}{a+\sin^2\theta}\), \(\int_{|z|=2}\frac{dz}{(z^5-1)(z-3)}\), \(\int_{-\infty}^{\infty}\frac{\sin(\pi a)\,e^{-ix\xi}}{\cosh(\pi x)+\cos(\pi a)}\,dx\), and \(\int_{|z|=1}\cot^2 z\,dz\)
- A \(C^1\) function with \(F(0,0)=0\) and \(\|\nabla F(0,0)\|<1\) satisfies \(|F|<r\) on some ball of radius \(r\)
- \(\int_{|z|=1}\bigl(z+\frac1z\bigr)^{2n}\frac{dz}{z}\) and \(\int_0^{2\pi}\cos^{2n}\theta\,d\theta=2\pi\frac{1\cdot 3\cdots(2n-1)}{2\cdot 4\cdots(2n)}\)
- \(z\mapsto\overline{f(\bar z)}\) is analytic if \(f\) is
- Compact-uniform limits of analytic functions on \(\DD\) are analytic
- Residue evaluations of seven real and contour integrals
- A nonvanishing holomorphic function with \(|f|=1\) on the unit circle is constant
- Geometrically describe the following subsets of \(\CC\): \(\abs{z-1} = 1\) \(\abs{z-1} = 2\abs{z-2}\) \(1/z = \bar z\) \(\Re(z) = 3\) \(\Im(z) = a\) with \(a\in \RR\).
- Analyticity of \(\Gamma(s)\) for \(\Re(s)>0\) and the reflection formula
- Uniform limits of continuous functions are continuous
- Polynomials in \(z\) are not uniformly dense in \(C(\overline{\mathbb{D}})\)
- Entire functions satisfying \(|f(z)|\geq|z|\) are the maps \(cz\) with \(|c|\geq 1\)
- Uniform differentiability on \([a,b]\) is equivalent to continuity of \(f'\)
- \(\int_\gamma\frac{f'(z)}{z-z_0}\,dz=\int_\gamma\frac{f(z)}{(z-z_0)^2}\,dz\)
- The number of solutions of \(6z^3+1=-e^z\) in \(|z|<1\)
- Equilateral triangles as \(z_1^2+z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1\)
- The sets \(|z-\alpha|+|z+\alpha|=k\) and \(|z-\alpha|+|z+\alpha|\leq k\) for \(2|\alpha|<k\)
- \(f(z)=b\) has \(n\) solutions in the unit disk for a product of Blaschke factors
- An entire function omitting a bounded open set is constant
- Singularities at infinity of entire functions with finitely or infinitely many zeros
- Invert \(1-z\), one coefficient at a time
- Polar Cauchy–Riemann equations and holomorphy of the principal logarithm
- Invert \(\sin(z)\) using geometric series
- Invert \(\sin(z)\), one coefficient at a time
- A holomorphic function with vanishing derivative on a connected domain is constant
- An uncountable \(E\subset[0,1]\) meets both \((-\infty,t)\) and \((t,\infty)\) in uncountable sets
- Arbitrary Rouché, \(R\mathbb{D}\)
- \(\int_0^\infty\frac{\sin^3 x}{x^3}\,dx\)
- Cauchy estimates for a holomorphic function of polynomial growth on a strip
- Uniform convergence of \(\sum\sin(nz)/2^n\) on \(\{\Im z<\ln 2\}\) and on \(|z|\le r\)
- Zeros of a polynomial in the closed unit disk
- Green’s theorem for rectangles, Cauchy’s theorem, and Goursat’s theorem
- equations take the form
- The Blaschke factor \(z\mapsto\frac{w-z}{1-\bar{w}z}\) as a biholomorphism of \(\DD\)
- A nowhere-vanishing holomorphic function on a simply connected domain is of the form \(e^g\)
- \(\frac{1}{2\pi i}\int_{|z|=2}\frac{z^n}{1-3z\cos\theta+z^2}\,dz=\frac{\sin(n\theta)}{\sin\theta}\)
- \(\lim a_n/a_{n+1}=z_0\) for the power series of a function with a pole at \(z_0\in\partial\DD\)
- A continuous \(f:\RR\to\RR\) vanishing at \(\pm\infty\) is uniformly continuous
- Entire functions satisfying \(|f(z)|\ge|z|\)
- Taylor coefficients of a function with a pole on the unit circle
- Zeros of \(P_n(z)=\sum_{k=0}^n z^k/k!\) and of \(P_n-1\) in \(|z|<10\) for large \(n\)
- Real differentiability: definition, partials without differentiability, and real but nowhere complex differentiability
- Entire functions converging uniformly on line segments converge uniformly on compacta
- Explicit Rouche, \(2\mathbb{D}\)
- \(\int_0^\infty\frac{x^{-s}}{x+1}\,dx=\frac{\pi}{\sin(\pi s)}\)
- An entire function with \(|f(z)|\leq M|z|^2\) outside a disk is a polynomial of degree at most \(2\)
- The sum \(\sum_{k=1}^n\sin kx\) via de Moivre’s theorem
- A holomorphic injection has nonvanishing derivative
- Real differentiability of \(f:\RR^2\to\RR\) via linear maps, the inverse function theorem, and the Cauchy–Riemann criterion for complex differentiability
- \(\int_0^\infty\frac{x^{a-1}}{1+x^n}\,dx=\frac{\pi}{n\sin(a\pi/n)}\) for \(0<a<n\)
- Holomorphic functions vanishing on the boundary are constant or have an interior zero
- The Fundamental Theorem of Algebra
- The Fourier transform of a compactly supported continuous real function is entire