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At an unramified irregular singular point with distinct leading eigenvalues, a sectorial fundamental solution is a fundamental matrix whose entries are holomorphic functions asymptotic to a prescribed formal matrix on a sector of the logarithm cover. For competing cubic exponentials, the usual construction uses six overlapping sectors with basic angle $\pi /3$. A formal power series need not converge, and two sectorial realizations may differ by exponentially small terms across a sector boundary.

The sheaf ${\mathcal{K}}_X$ assigns to an open subset of a complex manifold its meromorphic functions. Its subsheaf ${\mathcal{K}}_X^{\ast }$ consists of the nonzero meromorphic functions, while ${\mathcal{O}}_X^{\ast }$ consists of the nowhere-zero holomorphic functions.

On a simply connected domain $D$, a germ of a holomorphic function admitting analytic continuation along every path from its base point determines a unique holomorphic function $F:D\to \mathbb{C}$ with that initial germ of a holomorphic function. ^[ref]

Indeed, any two paths with the same endpoints are homotopic with endpoints fixed. Define $F(b)$ using the terminal germ of a holomorphic function. The monodromy theorem makes this independent of the path. In a small open disc about $b$, extend the chosen route by a short segment; the same local representative supplies $F$ throughout the open disc, proving holomorphy. The identity theorem proves uniqueness. The existence hypothesis of the monodromy theorem must still be checked.

Reality of the boundary derivatives gives $H_1(\bar{k})=\overline{H_1(k)}$ and $H_2(-\bar{k})=\overline{H_2(k)}$, and the same identities for $J_1,J_2$. The global relation and its conjugate express both quarter-plane oblique boundary spectral functions in terms of the known $H_1,H_2$ and the single bounded holomorphic function $\Phi$ in the upper half-plane. For $\gamma ={\beta }_1+{\beta }_2\in \{0,\pi /2,\pi \}$, its coefficients on the positive real and positive imaginary reconstruction rays are opposite, because $e^{4i\gamma }=1$. Closing the first-quadrant contour removes $\Phi$ by the Cauchy integral theorem.

// Target: analysis.bigb

For an open set $U\subseteq \mathbb{C}$, the space $\mathcal{O}(U)$ consists of all holomorphic functions on $U$. Its standard topology is locally uniform convergence, equivalently the compact-open topology defined by the seminorms $f\mapsto {\sup }_{z\in K}|f(z)|$ for compact $K\subset U$.

For a time Laplace transform parameter with $\mathrm{Re}p>0$, the spatial characteristic roots satisfy $r^3+r+p=0$. Exactly one root $r_{-}$ has negative real part and two have positive real part. Indeed an imaginary root would force $p=-i(k-k^3)$ to be imaginary. The count is therefore constant on the connected right half-plane; for real $p>0$, the cubic has one negative real root and a conjugate pair with positive real parts. Repeated roots would satisfy $3r^2+1=0$, which again puts $p$ on the imaginary axis. Thus $r_{-}(p)$ is a nonzero holomorphic function in the right half-plane. Only $e^{r_{-}x}$ decays at positive spatial infinity.

If a spectrum of an element splits into disjoint nonempty compact pieces $X,Y$, choose disjoint open neighborhoods and a holomorphic function equal to one on the first and zero on the second. The holomorphic functional calculus sends it to an idempotent with the corresponding zero/one values under every algebra character. It is neither zero nor identity: if one piece were killed, a reciprocal function on the other piece would make an originally spectral coordinate difference invertible.

For a Banach algebra-valued holomorphic function, set $p_n(w)=\Vert f(w{)}^{2^n}{\Vert }^{1/2^n}$. These continuous functions decrease to the spectral radius by the spectral radius formula. If $M={\sup }_{\partial K}r(f(w))$, the continuous functions $\max \{p_n,M\}$ on $\partial K$ decrease to the continuous constant $M$. The Dini theorem gives uniform convergence. Apply the norm maximum principle for a Banach-space-valued holomorphic function to each power, and then let $n\to \infty$. This proof does not assume continuity of spectral radius.

For subsonic motion $|V|<c$, an antiplane shear displacement $w(x_1-Vt,x_2)$ satisfies ${\beta }^2{w}_{xx}+w_{x_2{x}_2}=0$. The displayed rescaling makes it a harmonic function of $(x,y)$, so locally $w=\mathrm{Re}H(x+iy)$ for a holomorphic function $H$. Set $F=\beta H$. The Cauchy-Riemann equations then give $\mu F^{\prime }=\beta {\sigma }_{13}-i{\sigma }_{23}$. Keeping this potential normalization explicit avoids losing a factor of $\beta$ in the dynamic stress field. At sonic or supersonic speeds the elliptic harmonic reduction does not apply.

A complex manifold is Stein if its global holomorphic functions separate points, give local coordinates, and make it holomorphically convex: the holomorphic hull of every compact set is compact. Closed complex submanifolds of affine complex space are examples, including products of copies of $\mathbb{C}$ and ${\mathbb{C}}^{\ast }$. Cartan theorem B makes these spaces useful for acyclic covers in sheaf cohomology.

The structure sheaf assigns to an open set its ring of holomorphic functions. Its invertible sections form the sheaf ${\mathcal{O}}_M^{\ast }$ of nowhere-zero holomorphic functions.

The coefficient of $(z-z_0{)}^n$ in the Taylor series of a function at $z_0$. The displayed formula defines it whenever the indicated derivative exists. For a holomorphic function on a disk, the Cauchy integral formula also gives $a_n=(2\pi i{)}^{-1}\oint f(z)(z-z_0{)}^{-n-1}dz$ along a circle enclosing the center within that disk.

On a compact connected complex manifold, pair nonzero holomorphic sections of a holomorphic line bundle and its holomorphic dual vector bundle. Their product is a global holomorphic function, hence constant by the maximum modulus principle. The identity theorem makes their nonvanishing loci dense, so this constant is nonzero. Both sections are consequently nowhere zero and trivialize their bundles. Conversely, the constant section of a trivial bundle and its dual provide the two sections.

For a holomorphic function on the unit disc, this boundary condition characterizes Blaschke products, including unimodular constants. The nondecreasing radial means of the nonnegative subharmonic function ${\log }^{+}|f|$ must vanish, so $|f|\le 1$. In the Blaschke factorization of a bounded holomorphic function $f=Bg$, both logarithmic means tend to zero. The mean value property for harmonic functions applied to $\log |g|$ gives $|g(0)|=1$, and the maximum modulus principle makes $g$ unimodular constant. The converse follows from the radial logarithmic mean of a Blaschke product. Omitting the outer absolute value changes the condition: $e^z$ has zero mean logarithmic modulus, but is not a Blaschke product.

A map from an open subset of $\mathbb{C}$ into a complex Banach space is weakly holomorphic when every composition with a bounded complex-linear functional is a scalar holomorphic function. Norm holomorphy requires its difference quotients to converge in the Banach norm. Weak holomorphy is initially a condition on scalar observations; the weak holomorphy implies norm holomorphy argument recovers an actual vector derivative. A closed target subspace containing values on a sequence with an interior accumulation point contains every value on the connected domain: annihilate the subspace, apply the scalar identity theorem, and use Hahn-Banach separation.

Here a weak modular form is a holomorphic function on the complex upper half-plane satisfying the weight-$k$ transformation law, with no growth condition at any cusp of a modular group. This convention differs from weakly holomorphic modular forms, which are required to be meromorphic at the cusps. At level one, translation invariance gives a Laurent expansion on the punctured unit disc, possibly with infinitely many negative powers.

The boundary can recover more than a count. Under the hypotheses of the argument principle, let $h$ be a holomorphic function on a neighborhood of the contour and its interior. If the enclosed zeros of a function are $a_j$ with zero multiplicities $m_j$ and the enclosed poles are $b_k$ of orders $n_k$, thenMultiplying the local logarithmic derivative by $h$ changes its residue from $m_j$ to $m_{j}h(a_j)$, or from $-n_k$ to $-n_{k}h(b_k)$. The residue theorem then gives the identity. Choosing $h=1$ recovers the argument principle; choosing $h(z)=z$ gives the sum of the enclosed zero locations minus the sum of the enclosed pole locations, both counted with zero multiplicities or pole orders.^[ref]

For the rational function in argument principle for a rational function, the $h(z)=z$ integral is $2(1/4)-(-1/2)=1$. Adding the known pole location $-1/2$ back gives a zero-location sum of $1/2$. Dividing by the zero count two gives the arithmetic mean of the enclosed zeros, with multiplicity, as $1/4$. Choosing $h(z)=z^k$ similarly gives the difference between the power-sum symmetric polynomials evaluated on the enclosed zero and pole locations, repeating each location according to its multiplicity or order. These identities let boundary information constrain both the number and aggregate locations of enclosed zeros of a function.

Let $R,\alpha >0$ and ${\mathcal{C}}_{\alpha }=\{(x,t)\in {\mathbb{C}}^2:|x|<R,|t|<\alpha (1-|x|/R)\}$. The displayed norm makes the holomorphic functions with finite norm a Banach space. Finiteness forces $u(x,0)=0$: for fixed $x$, the weight grows like $1/|t|$ near zero. A Cauchy sequence in this norm converges uniformly on every compact subset of ${\mathcal{C}}_{\alpha }$, including at $t=0$. Its limit is holomorphic by locally uniform convergence of holomorphic functions, and the pointwise norm bounds show convergence in the original norm. The shrinking domain compensates for loss of an $x$-derivative in a Cauchy-Kovalevskaya theorem argument.

For $z=x+iy$, the Wirtinger derivatives are ${\partial }_z=({\partial }_x-i{\partial }_y)/2$ and ${\partial }_{\bar{z}}=({\partial }_x+i{\partial }_y)/2$. The Cauchy-Riemann equations become ${\partial }_{\bar{z}}F=0$ for a holomorphic function $F$, and $4{\partial }_z{\partial }_{\bar{z}}=\Delta$. Thus $u_z$ is holomorphic whenever $u$ is a twice continuously differentiable harmonic function.

For a zero-free holomorphic function $f:\mathbb{D}\to \mathbb{D}$, put $c=f(0)$ and $\alpha =-\log |c|$. A holomorphic logarithm makes $-\log f$ have positive real part. The Carathéodory derivative bound for the right half-plane yields the displayed sharp bound. Equality functions are $f(z)=c\exp [-2\alpha \omega z/(1-\omega z)]$, $|\omega |=1$. The Harnack inequality on the unit disk also yields the sharp pointwise bounds $|c{|}^{(1+r)/(1-r)}\le |f(w)|\le |c{|}^{(1-r)/(1+r)}$ for $r=|w|$. Aligning $\omega w=\pm r$ attains the two endpoints.