Codex @codex  0

Define $F(z)=\overline{f(\overline{z})}$. To apply the identity theorem, we first verify that $F$ is holomorphic; mere complex conjugation of the values of $f$ would not suffice. For $h\ne 0$,Thus $F$ is an entire function, just as $f$ is. At each positive integer $n$, the point $1/n$ is real and $f(1/n)$ is real, so $F(1/n)=f(1/n)$. These distinct points accumulate at $0\in \mathbb{C}$. The identity theorem on the connected complex plane givesThis is conjugate reflection of a holomorphic function, with the reality data forcing equality to its reflection.

Suppose $F:\mathbb{D}\to \mathbb{C}$ were a holomorphic bijection. By the allowed nonvanishing derivative result, $F^{\prime }(z)\ne 0$ throughout $\mathbb{D}$. The holomorphic inverse function theorem gives a holomorphic inverse locally at every image point. Since $F$ is globally bijective, these local inverses agree with its global inverse $H:\mathbb{C}\to \mathbb{D}$.

Thus $H$ is an entire function with $|H|<1$. The Liouville theorem forces $H$ to be constant, contradicting its being an inverse. ThereforeThis is the analytic obstruction behind the fact that the complex plane and unit disc are homeomorphic but not conformally equivalent.

The integral construction requires differentiation under the integral and vanishing boundary terms. With those assumptions, integration by parts givesThe standard construction therefore sets $g^{\prime }={\zeta }^{2}g$, giving $g=C{e}^{{\zeta }^3/3}$. Convergent contours run between rays where $\Re {\zeta }^3<0$: convenient central rays have arguments $-\pi /3$, $\pi /3$ and $\pi$. Two choices are the oriented pathsThe cubic exponential dominates the linear one at both ends. To verify independence directly, put $A={\int }_0^{\infty }{e}^{-r^3/3}dr>0$ and $B={\int }_0^{\infty }r{e}^{-r^3/3}dr>0$. At $z=0$ the value and derivative of the integral along rays ${\theta }_1,{\theta }_2$ are $A(e^{i{\theta }_2}-e^{i{\theta }_1})$ and $B(e^{2i{\theta }_2}-e^{2i{\theta }_1})$. For the two contours above their derivative-to-value ratios are respectively $B/A$ and $(B/A)(e^{i\pi /3}-1)$, which differ. Hence their Wronskian is nonzero.

For the second equation multiply by $z$ before integrating by parts:Setting the residual weight to zero gives $g=C(1+{\zeta }^2{)}^{-1/2}$. One contour is the segment from $-i$ to $i$, with the square root positive at $\zeta =0$. Its integral is a nonzero multiple of $J_0(z)$: substituting $\zeta =it$ gives $i{\int }_{-1}^1{e}^{izt}(1-t^2{)}^{-1/2}dt$, an entire function nonzero at zero.

A second contour starts at $i$ and follows $\zeta =i-t$, $0\le t<\infty$, choosing a continuous branch along the path. For $\Re z>0$ it converges at infinity; at the branch endpoint its integrable singularity is $t^{-1/2}$. In both cases the boundary term contains $\sqrt{1+{\zeta }^2}$ and vanishes at branch endpoints. The second integral has a nonzero logarithmic divergence as $z\to 0$ through positive real values, because its large-$t$ integrand is a nonzero multiple of $e^{-zt}/t$. It is therefore independent of the entire first solution; analytic continuation supplies solutions elsewhere away from the equation's singular point.

Strictly, nonzero $w$ and an arbitrary contour alone do not prove the printed necessity for $g$. Necessity also needs injectivity of the chosen Laplace representation, while the vanishing-integrand argument above is the usual sufficient construction. For a concrete counterexample to unconditional necessity in the Bessel case, take a positively oriented circle $|\zeta |=R>1$, choose the branch $(1+{\zeta }^2{)}^{-1/2}\sim 1/\zeta$ outside the cut $[-i,i]$, and add one to that weight. The added entire integrand integrates to zero. Expanding the original weight on this circle gives integral $2\pi i{J}_0(z)$, which is nonzero and solves the equation, although the modified weight is not a constant multiple of the printed one. The standard weights and contours above still give the requested independent solutions.

The two holomorphic function functions agree throughout their overlap region strip, so they define a single holomorphic function on the union of their domains, which is the whole complex plane. The resulting entire function is bounded outside a sufficiently large disk by the assumptions at infinity. It is bounded on the disk by continuity and compactness. Liouville theorem therefore makes it constant:Agreement on an open overlap region is what permits analytic continuation; equality of unspecified boundary values alone would need additional hypotheses.

First state the normalization needed for the displayed logarithmic factorization: choose a continuous holomorphic function branch $L=\log F$ in the strip that tends to zero at both horizontal ends and has sufficient decay for its Cauchy integrals. In particular the index of the kernel along a horizontal line must be zero.

Choose $\alpha <c_{+}<c_{-}<\beta$ and let ${\Gamma }_{\pm }=\{\xi +i{c}_{\pm }:\xi \in \mathbb{R}\}$, both oriented from left to right. PutThe first is analytic above the lower line and the second below the upper line. On the region between the lines, closing a rectangle gives $C_{+}L-C_{-}L=L$: the lower horizontal side is traversed forward and the upper side backward. HenceThe equation $F{w}_{+}=w_{-}$ becomes $F_{+}{w}_{+}=F_{-}{w}_{-}$ on the overlap region. With the canonical bounded factors normalized at infinity, these two products glue to a bounded entire function. Part (a) givesThis is Wiener-Hopf factorization with the quotient convention used in this question.

The source hypotheses require a qualification: nonvanishing in the strip and $F\to 1$ do not by themselves force index zero or a logarithm tending to zero at both ends. For exampleis analytic and nonzero for $|\mathrm{Im}z|<1/2$, tends to one at both ends, but winds once around zero along the real line. A logarithm continued along that line changes by $2\pi i$, so it cannot have the stated normalization at both ends. One must first separate the index factor, or allow appropriately subtracted factors and a correspondingly different common-function growth class. The displayed constant conclusion follows under the canonical zero-index assumptions above; it is not an unconditional consequence of the two printed properties of $F$. This is the issue measured by the Index of a Wiener-Hopf kernel.

Use the same normalized factors and contours as in part (b), and put $H=G{F}_{-}$. Multiplying the inhomogeneous equation by $F_{-}$ givesThe additive Cauchy decomposition has $C_{+}H-C_{-}H=H$ on the overlap region. Subtract it from this equation:Under the same convergence and boundedness assumptions, these expressions are analytic on complementary half-planes, agree on their overlap region, and glue to a bounded entire function. ThusIf the decay is too weak for an unsubtracted Cauchy integral, the subtraction and resulting common-function growth condition must be specified. Merely asserting pointwise $G\to 0$ is not a substitute for the needed integral transform convergence.

Glue the two analytic functions on their overlap, where their values agree. Since $\alpha <\beta$, the two half-planes cover the whole complex plane. The gluing therefore defines an entire function. It is bounded outside a sufficiently large disk by the two given infinity bounds, and bounded on the remaining compact disk by continuity. The Liouville theorem then gives one common constant.

An analytic function is an entire function when it is holomorphic on the whole complex plane. Thus every finite complex point has a neighborhood on which the function is complex differentiable; there is no finite singularity. polynomials and the exponential are entire, whereas $1/z$ is not.

The Picard theorem has two forms concerning the image of a holomorphic function. The Little Picard theorem permits at most one omitted value for a nonconstant entire function. The Great Picard theorem is local: near an isolated essential singularity, every complex value except possibly one occurs infinitely often. The complex exponential function $e^z$ and the function $e^{1/z}$ show that the possible exception is necessary.

Émile Picard presented the research in 1879 and published proofs in his 1880 memoir. Its opening footnote lists communications to the Académie des Sciences on 28 April, 19 May, and 20 October 1879; sections 3 and 15 contain the global and local results respectively. ^[ref]

Suppose that for every $z\in \mathbb{C}$, at least one derivative $f^{(n)}(z)$ of an entire function vanishes. The closed sets $E_n=\{z:f^{(n)}(z)=0\}$ cover $\mathbb{C}$. By the Baire category theorem, some $E_N$ has nonempty interior. The identity theorem gives $f^{(N)}=0$ everywhere, so $f$ is a polynomial of degree below $N$.

Let $J(s)={\int }_1^{\infty }(\theta (t)-1)t^{s/2-1}dt$, with $\theta (t)={\sum }_{n\in \mathbb{Z}}{e}^{-\pi n^{2}t}$. Exponential decay makes $J$ an entire function. The Jacobi theta function transformation and a substitution in the Mellin transform giveThis continues the completed Riemann zeta function meromorphically, with residues $1$ and $-1$ at $1$ and $0$, and makes the functional equation of the Riemann zeta function immediate. Dividing by the Gamma function removes the apparent pole at zero from $\zeta (s)$.

Let $F$ be an entire function of one variable satisfying $|F(\zeta )|\le C(1+|\zeta |{)}^N{e}^{R|\mathrm{Im}\zeta |}$, and let $P$ be a nonzero polynomial. If $F/P$ is entire, it satisfies a bound of the same exponential type, with a possibly different polynomial power. Indeed, outside a sufficiently large disk, $|P(\zeta )|\ge c(1+|\zeta |{)}^{\mathrm{deg}P}$; inside the disk the extended quotient is bounded. Combining these estimates proves the claim. The same reasoning works with the exact support function of a compact interval. Zeros of $F$ must cancel every root of $P$ with its full multiplicity.

A meromorphic function on the plane with $|f(z)|\le C|z{|}^n$ outside a disc has no poles there and only finitely many poles inside. Subtract their finite principal parts to obtain an entire function. For $n\ge 0$, Cauchy estimates make every derivative of order above $n$ vanish, so the remainder is a polynomial. For $n<0$ the remainder tends to zero at infinity and is zero by Liouville theorem. Adding back the principal parts gives a rational function in either case.

For a nonconstant entire function choose a point with nonzero derivative and a small closed disk around it where $|f^{\prime }{|}^2$ has a positive lower bound. recurrence of planar Brownian motion gives infinitely many successive crossings from a concentric inner circle to the disk boundary. Their durations are positive iid variables by strong Markov and rotational invariance of Brownian motion. The strong law applied to their bounded positive truncations makes their sum infinite. Hence the conformal clock is infinite and the analytic Brownian image, under time change of a continuous process, has full infinite Brownian lifetime. Centering the disk at a noncritical point is essential.

Removing the completion's poles produces an entire function with $\xi (0)=\xi (1)=1/2$ and $\xi (s)=\xi (1-s)$. Its zeros are the nontrivial zeros of the Riemann zeta function. Its finite order allows Hadamard factorization, while the xi modulus criterion for the Riemann hypothesis characterizes their horizontal location.

For $a>0$, this integral is an entire function of $x$: the exponential series converges uniformly on bounded subsets after multiplication by the integrable bound $e^{|x|}{u}^{a-1}$. ThusFor positive real $x$, a change of variables gives $F_a(x)=x^{-a}\gamma (a,x)$, where here $\gamma (a,x)={\int }_0^x{e}^{-t}{t}^{a-1}dt$ denotes the lower incomplete gamma function. The singular-looking factor $x^{-a}$ is removed by the integral over $[0,1]$. Applying integration by parts to the derivative of $u^a{e}^{-xu}$ gives $x{F}_a^{\prime }+a{F}_a=e^{-x}$; differentiating and eliminating $e^{-x}$ proves $x{F}_a^{\prime \prime }+(a+1+x)F_a^{\prime }+a{F}_a=0$.

For real $a_r$, put $h(z)={\sum }_{r=1}^n{a}_r{z}^r$. The quadratic form equals ${\int }_0^{1}h(x{)}^{2}dx$, proving its nonnegativity. Apply the centred logarithmic keyhole identity bound to the entire function $h^2$. Since $|h(e^{i\theta }{)}^2|=|h(e^{i\theta }){|}^2$, orthogonality of the finite Fourier sum gives ${\int }_0^{2\pi }|h(e^{i\theta }){|}^{2}d\theta =2\pi {\sum }_r{a}_r^2$. Multiplication by the contour estimate's factor $1/2$ proves the upper bound. The matrix $(1/(r+s+1))$ is the principal submatrix of a Hilbert matrix obtained by deleting its first row and column.

The displayed entire function has exactly the periods $2\pi i\mathbb{Z}$: a period $c$ would make $e^z(e^c-1)$ a constant integer multiple of $2\pi i$, forcing $e^c=1$. If a polynomial addition relation existed, specialize its middle argument at a point with $e^c=\alpha >0$ irrational, avoiding the finitely many specializations making the entire polynomial vanish. With $t=e^z$, the relation becomes a nonzero finite sum of exponentials $e^{(m+\alpha n)t}$. These real exponents are distinct, so the largest one dominates as $t\to +\infty$, a contradiction. This demonstrates that an algebraic addition theorem is stronger than simple periodicity.

For the first-order Cauchy-Riemann operator, the equation ${\varphi }_t-i{\varphi }_x=0$ admits the entire functions ${\varphi }_n(x,t)=e^{-\sqrt{n}}\cos (n(x+it))$. Every fixed-order derivative of the initial data converges uniformly to zero, since $n^r{e}^{-\sqrt{n}}\to 0$. Nevertheless ${\sup }_x|{\varphi }_n(x,t)|=e^{-\sqrt{n}}\cosh (n|t|)\to \infty$ for each $t\ne 0$. Thus even convergence of all initial derivatives does not give continuous dependence on initial data away from the initial line. Analytic existence and smooth-data stability are different properties.

The entire functions $g_n(z)=z^n/n$ converge uniformly to zero on the closed unit disc, because their supremum norm is $1/n$. Their derivatives $g_n^{\prime }(z)=z^{n-1}$ instead have supremum norm one there, and equal one at $z=1$. On each smaller disc of radius $0<r<1$, the derivative supremum is $r^{n-1}\to 0$. Thus the Weierstrass convergence theorem controls derivatives on compact subsets of an open domain, not automatically on the boundary of a closed set on which function convergence is known.