# Questions tagged [entire-functions]

This tag is for questions relating to the questions on entire functions. The polynomials which form a special and important class of entire functions, can be characterized as those entire function which have at most a pole as a singularity at infinity.

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### Suppose $f_1,f_2$ entire functions. Produce entire $h,g_1,g_2$ such that $f_1=hg_1$ and $f_2=hg_2$ with $g_1,g_2$ no common zeros.

Suppose $f_1,f_2$ entire functions. Produce entire $h,g_1,g_2$ such that $f_1=hg_1$ and $f_2=hg_2$ with $g_1,g_2$ no common zeros. I know I have to use Weierstrass factorization theorem somehow but I’...
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### Maximum value of an entire function in a closed set.

Consider the entire function $f(z)=z(z-i)$. Put $$S=\bigg\{\frac{1}{|f(z)|}\ |\ |z|\geq 2\bigg\}.$$ At what value(s) of $z$ is the maximum of the set $S$ attained? My Idea: As $f(z)$ is an entire ...
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### Proving that the entire function which satisfies a given property is unique

Let $\phi : \mathbb{C} \to \mathbb{C}$ be an entire function satisfying the following three properties: $|\phi'(z)| \leq |\phi(z)|$ for all $z \in \mathbb{C}$ $\phi(0) = 2$ $\phi(1) = 1$ The problem ...
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Consider $f$ an entire function that $\int_{|z|=1}\frac{f(z)}{((k+1)z-1)^n}dz=0$ for any $k\in\Bbb N$. What can we say about $f$ for $n\in\{ 1,2,3,\cdots\}$? We know that $\frac{n!}{2\pi i}\int_{|z|=... 1 vote 1 answer 65 views ### How to deduce that there doesn't exist any sequence of polynomials$P_n$such that$P_n(z)\to \frac 1z$The exercise I have reads as follow: Let$f$be an entire function. Compute \begin{equation} \frac1{2\pi i}\int_{|z|=1}\frac{1-zf(z)}{z}dz \end{equation} Prove that all$f$entire functions verify ... 0 votes 1 answer 61 views ### If$f$is an entire function,$\lim_{z\to\infty} z^{-1}\operatorname{ Re} f(z)=0$, show$f$is constant. If$f$is an entire function,$\lim_{z\to\infty} z^{-1} \operatorname{ Re} f(z)=0$, show$f$is constant. I know only the Liouville theorem: bounded entire funcion is constant. It seems that the ... 0 votes 2 answers 55 views ### What can we say about these two entire functions? Let$g$and$h$be two entire functions such that$\lim_{|z|\to\infty}\frac{|g(z)|}{|z|^5} = 0$and$|h(z)|\le |z|^4$. What can we say about$g$and$h$? I got to both of them being polynomials of at ... 0 votes 0 answers 52 views ### Is a function with an essential singularity always some composition of an entire transcendental (not a polynomial) function? I thought about the theorem of Casorati-Weierstraß(in my lecture notes there are actually two, one deals with entire functions and the other with essential singularities) and asked myself if the ... 1 vote 0 answers 124 views ### If$ |f(z^{2})|\leq|f(z)| $and$ f $is entire, then is$ f $constant? [duplicate] Assume$f:\mathbb{C}\to \mathbb{C} $entire function, and assume$ f $satisfies$ |f(z^{2})|\leq|f(z)| $How can I prove that$ f $is a constant? I have tried something but not sure about my way. I'... 0 votes 0 answers 72 views ### Prove that$f(z) = f(−z)∀ z ∈ \Bbb C$Let f be an entire function such that$f (ix) = f (x)$for all$x ∈ (1, 2) ⊆ \Bbb R$. Prove that$f(z) = f(−z)∀ z ∈ \Bbb C$I know zero sets of non-constant holomorphic functions are discrete, i.e. ... 0 votes 0 answers 34 views ### show that an entire function is constant [duplicate] Let f be an entire function with the property that$ |f(\frac{1}{n})|< n^{-n}$for$n ∈ \Bbb N$. Show that f is constant I know it follows from Liouville Theorem that if an entire function is ... 0 votes 1 answer 84 views ### Does there exist$z_0 \notin \{a_n\}_{n \geq 1}$such that$f(z_0) = 0\ $? I am reading Weierstrass Factorization Theorem from the lecture notes given by our instructor. Here I came across a theorem which is the following$:\mathbf {Theorem} :$Let$\{a_n\}_{n \geq 1}$... 2 votes 1 answer 91 views ###$f,g$entire such that$f(0)=g(0)\neq 0$and$|f(z)|\leq |g(z)|$for all$z\in\mathbb{C}$, then$f=g$. Question: Suppose we have functions$f,g$entire such that$f(0)=g(0)\neq 0$and$|f(z)|\leq |g(z)|$for all$z\in\mathbb{C}$, then$f=g$. My attempt: Consider function$h(z)=\frac{f(z)}{g(z)}$, ... 0 votes 1 answer 54 views ### Do Meromorphic Functions automatically give us Entire Functions? Suppose I have a meromorphic function,$f$. Then, I can write$f(z)=\frac{h(z)}{g(z)}$where$h,g$are entire. I would really like to be able to claim that$f$extends to an entire function by ... 1 vote 0 answers 45 views ### If$f$is an non-null entire function that satisfies$|f(z)|=2$for all$z\in\partial \Bbb D$, then$f$is constant? [duplicate] I want to prove if either this statement is true or false: Let$f:\Bbb C\to \Bbb C$be a non-null entire function which verifies that$|f(z)|=2$for every$z$that belongs in the circle of center the ... 3 votes 1 answer 116 views ###$\lim_{\vert z\vert\to\infty}\frac{zf'(z)}{f(z)}=n\in\mathbb{N}$implies$f$is a polynomial Let$f$be an entire function, meaning $$f:\mathbb{C}\to\mathbb{C}$$ is holomorphic. If$f\not\equiv0$and $$\lim_{\vert z\vert\to\infty}\frac{zf'(z)}{f(z)}=n\in\mathbb{N}_0$$ then$f$have to be a ... 1 vote 1 answer 166 views ### Order of growth of$\prod_{n=1}^{\infty}(1-a^nz)$for$0<|a|<1$This question is from Conway Complex Analysis, page 287, exercise 9(a). My attempt: Write the product as$\underset{n}\prod(1-\frac{z}{b^n})$, where$b=1/a$. First note that this entire function has ... 0 votes 1 answer 114 views ### Which of the following functions is/are constant? Which of the following functions is/are constant ? Let$f(z)$be an analytic function in extended complex plane. If$g(z)$is an entire function such that$g(z) = u + i v$and$u^2 \leq v^2 + 2012$... 2 votes 1 answer 126 views ### Show that$\int_1^{\infty} t^{x-1} e^{-t} dt\$ is entire. 