Intereting Posts

The partial fraction expansion of $\frac{1}{x^n – 1}$.
Prove that if Rank$(A)=n$, then Rank$(AB)=$Rank$(B)$
A partition of the unit interval into uncountably many dense uncountable subsets
Find $\det X$ if $8GX=XX^T$
Distribution for random harmonic series
A question related to Pigeonhole Principle
$ z^n = a_n + b_ni $ Show that $ b_{n+2} – 2b_{n+1} + 5b_n = 0 $ (complex numbers)
How to show this sequence is a delta sequence?
Smallest non-commutative ring with unity
why symmetric matrices are diagonalizable?
Maximum of the sum of cube
How to prove: $\sum_{k=m+1}^{n} (-1)^{k} \binom{n}{k}\binom{k-1}{m}= (-1)^{m+1}$
What is a Real Number?
Induction Inequality Proof with Product Operator $\prod_{k=1}^{n} \frac{(2k-1)}{2k} \leq \frac{1}{\sqrt{3k+1}}$
Can you help me understand this definition for the limit of a sequence?

Suppose $h(z)$ is a complex function. I have noticed that $h(z) = f(z)\cdot g(z)$, where $f(z)$ and $g(z)$ are non-holomorphic. Can $h(z)$ be holomorphic?

Can a similar statement be made, if $f(z)$ is holomorphic, but $g(z)$ is not?

- show that function $f(z)=(z^2+3z)/(e^z-1)$ can be expressed as a power series
- Some inequalities for an entire function $f$
- Fourier transform of $f(x)=\frac{1}{e^x+e^{-x}+2}$
- Removable singularities and an entire function
- solve $\sin(z)=-1$ in the set of complex numbers
- Every Cauchy sequence in $\mathbb{C}$ is bounded

- Showing equality of sequence of holomorphic functions to limit function if converges uniformly locally
- The simple roots of a polynomial are smooth functions with respect to the coefficients of the polynomial?
- Define a Principal Branch in Complex Analysis
- Show that $f_k'(z)=z^{k-1}$ does not converge uniformly for $|z|<1$.
- How to express $z^8 − 1$ as the product of two linear factors and three quadratic factors
- If a holomorphic function $f$ has modulus $1$ on the unit circle, why does $f(z_0)=0$ for some $z_0$ in the disk?
- Complex number condition on the modulus
- on the boundary of analytic functions
- Integral $\int_0^\infty \log(1+x^2)\frac{\cosh \pi x +\pi x\sinh \pi x}{\cosh^2 \pi x}\frac{dx}{x^2}=4-\pi$?
- Why there is no continuous argument function on $\mathbb{C}\setminus\{0\}$?

Yes, consider $\bar z \cdot \dfrac 1{\bar z} = 1$. However, if $f$ is holomorphic and $h$ is holomorphic, then $g=h/f$ is holomorphic wherever $f\ne 0$.

Most of the time, you can make a similar statement when $f$ is holomorphic while $g$ is not, but you have to modify it a bit to rule out silly cases like $f(z)=z^2$ and $g(z)=\frac1z$ when $z\neq0$ and $g(z)=0$. In this case, $h(z)=z$ is holomorphic, but in reasonable cases, $h$ will not be holomorphic.

Specifically, When $f$ is holomorphic and $g$ is not *meromorphic*, you can conclude have $h$ is not holomorphic.

Assume $h$ is holomorphic. Since $1/f$ is holomorphic except at the zeroes of $f$, which must be isolated, we have that $h\cdot(1/f)=g$ is holomorphic except for some poles or removable singularities (depending on whether the zeroes of $h$ cancel those of $f$). This shows $g$ is meromorphic.

- On Fredholm operator on Hilbert spaces
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- Stuck trying to prove an inequality
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- Proving Holder's inequality using Jensen's inequality
- Branch of math studying relations
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- $G$ is $p$-supersoluble iff $|G : M | = p$ for each maximal subgroup $M$ in $G$ with $p | |G : M |$.
- Continuously extending a set of independent vectors to a basis.
- Looking for an example of an infinite metric space $X$ such that there exist a continuous bijection $f: X \to X$ which is not a homeomorphism
- Integral involving a dilogarithm versus an Euler sum.