Intereting Posts

An inequality regarding the derivative of two concave functions
Why are the solutions of polynomial equations so unconstrained over the quaternions?
How to get inverse of formula for sum of integers from 1 to n?
All models of $\mathsf{ZFC}$ between $V$ and $V$ are generic extensions of $V$
Mean Value property for harmonic functions on regions other than balls/spheres
Probability of Obtaining the Roots in a Quadratic Equation by Throwing a Die Three Times
How do you solve the Initial value probelm $dp/dt = 10p(1-p), p(0)=0.1$?
Dimension of $GL(n, \mathbb{R})$
A 3rd grade math problem: fill in blanks with numbers to obtain a valid equation
How do I convert a fraction in base 10 to a quad fraction (base 4)?
What is the reason behind the current Order of Operations? (PEMDAS)
Question about normal subgroup and relatively prime index
Linear Algebra, eigenvalues and eigenvectors
how to find the root of permutation
Characterizing superposition of two renewal processes

For each fixed $n$, show that $$f_n(z)=\int_1^nt^{z-1}e^{-t}dt$$ is an

entire function of $z$.

From Morera ‘s theorem:

If a continuous, complex-valued function $f$ in a domain $D$ that

satisfies

- Residue at essential singularity
- Partial fractions for $\pi \cot(\pi z)$
- Residue of $f(z) = \frac{z}{1-\cos(z)}$ at $z=0$
- $|e^a-e^b| \leq |a-b|$
- An entire function $f(z)$ is real iff $z$ is real
- Representation of Holomorphic Functions By Exponential
$$\oint_\gamma f(z)\,dz = 0 $$

for every closed contour in $D$, then $f$ is analytic.

From this theorem how can I show that my function is indeed continuous? Do I prove this straight from the definition of continuity?

A hint which was provided, which I don’t understand why, goes:$$\text{Look at:} \ |f(z_1)-f(z_2)|\le \int_1^n\frac{e^{-t}}{t}|t^{z_1}-t^{z_2}|dt \ \text{where},\\ z_1=x_1+iy_1 \\ z_2=x_2+iy_2$$

thus I must bound this but I don’t know how to go from here? Any hints please?

- Find and sketch the image of the straight line $z = (1+ia)t+ib$ under the map $w=e^{z}$
- Operational details (Implementation) of Kneser's method of fractional iteration of function $\exp(x)$?
- Tricky contour integral resulting from the integration of $\sin ax / (x^2+b^2)$ over the positive halfline
- Is a pathwise-continuous function continuous?
- Proving $-\frac{1}{2}(z+\frac{1}{z})$ maps upper half disk onto upper half plane
- Integral $\int_0^{\infty} \frac{\ln \cos^2 x}{x^2}dx=-\pi$
- identity of $(I-z^nT^n)^{-1} =\frac{1}{n}$
- Liouville's theorem for subharmonic functions
- “Orientation” of $\zeta$ zeroes on the critical line.
- Recursion relation for Euler numbers

Related problems: I. Just see this,

$$ \oint_\gamma f_n(z) dz = \oint_\gamma \int_1^nt^{z-1}e^{-t}dt dz = \int_1^n e^{-t}\frac{dt}{t}\oint_\gamma e^{z\ln t} dz \,dt = \dots. $$

Can you finish it. What kind of function is $e^{z\ln(t)}$?

**Added:** To prove continuity, advance like that

$$ |f_n(z+h)-f_n(z)| = \Big|\int_1^n (e^{(z+h)\ln t}-e^{ z \ln t }) \frac{e^{-t}}{t}dt \Big| $$

$$ \leq \int_1^n \big|e^{(z+h)\ln t}-e^{ z \ln t }\big| \frac{e^{-t}}{t}dt \leq M e^{-1} |h|\int_{1}^{n} \ln t\frac{1}{t}dt\dots. $$

Can you finish the task? Notice that, I used the mean value theorem to find the last inequality and $ e^{-t} \leq e^{-1} $ for $t\in [1,n]$.

- Direct sum of orthogonal subspaces
- $\varphi\colon M\to N$ continuous and open. Then $f$ continuous iff $f\circ\varphi$ continuous.
- Why is gradient noise better quality than value noise?
- Does there exist $\alpha \in \mathbb{R}$ and a field $F \subset \mathbb{R} $ such that $F(\alpha)=\mathbb{R}$?
- Constructing the 11-gon by splitting an angle in five
- Why is $f(x) = x\phi(x)$ one-to-one?
- Elementary manipulation with elements of group
- Why is this weaker then Uniform Integrability?
- Radius of convergence of entire function
- how to calculate what I need for final exam
- Proving that the general linear group is a differentiable manifold
- If $\gcd(a,b)=1$, $\gcd(a,y)=1$ and $\gcd(b,x)=1$ then prove that $ax+by$ is prime to $ab$
- Sum of independent Gamma distributions is a Gamma distribution
- understanding $\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1$
- Prove that $p$ divides $F_{p-1}+F_{p+1}-1$