Eigenvalue problem for ODE in complex plane February 12

Eigenvalue problem for ODE in complex plane
To find the eigenvalue in the complex plane of $x$ for one dimensional Schrodinger equation $$ −ψ''(x) − (ix)^ N ψ(x) = Eψ(x). $$ where $N$ can be any real number, the boundary condition $ψ( ...

Complex 3-D Euclidean spaceinner product February 12    1

Complex 3-D Euclidean spaceinner product
1st question: Lets say we have a 3-D complex euclidean space. How do we geometrically draw this space? if 3-D real Euclidean space is represented by these base vectors:2nd question: Is there ...

Fourier Series with Complex Exponentials February 12    3

In my Signals and Systems class, we learned that the Fourier Series of a signal $x(t)$ is given by$$ x(t) = \sum_{k = -\infty}^{\infty} X_k e^{ik\omega_0t} $$where $\omega_0 = 2\pi/p$ and$$ X_k = \frac{1}{p} \int_0^p x(t) e^{-ik\omega_0t} \, dt. $$I

Calculate limit $(\frac{2x+1}{x-1})^x$ as $x$ goes to $\infty$ February 12    2

I have to calculate the following limit:$$\lim_{x\rightarrow \infty} \left(\frac{2x+1}{x-1} \right)^x$$$$\lim_{x\rightarrow \infty} \left( \frac{2x+1}{x-1} \right)^x=\lim_{x\rightarrow \infty} \left(1+\frac{x+2}{x-1} \right)^x=\infty$$But is $2^\inft

Complex euclidean tensor products February 12

Say you have Euclidean vectors $\mathbf{a}=a_i \mathbf{p}_i$ and $\mathbf{b}=b_j \mathbf{q}_j$ in $\mathbb{R}^3$, with bases $\mathbf{p}_i$ and $\mathbf{q}_j$. Then you could use a typical inner product to find $\mathbf{a}\cdot\mathbf{b}=a_i b_j (\ma

Hatcher deduce ring structure on $\mathbb{R}P^\infty;\mathbb{Z}$ February 12

Hatcher deduce ring structure on $\mathbb{R}P^\infty;\mathbb{Z}$
So in Hatcher they deduce the ring structure of $H^\ast (\mathbb{R}P^\infty;\mathbb{Z})$ by looking at the map $\mathbb{Z}\rightarrow\mathbb{Z}_2$, which induces maps on $H^\ast(\mathbb{R}P^ ...

Exponential of a complex variable February 12    1

Can someone please tell me if I am approaching this correctly? Given the following and asked to solve for the complex variable z: $$[e^z]^e^z=0$$My approach was purely algebraic and is why I have my doubts: $$[e^z]^3=5e^z$$ $$[e^z]^2=5$$ $$z=\frac

Eigenvalues of matrix summation February 12    1

Let $A$ be symmetric positive definite matrix with eigenvalues $\lambda_1,\lambda_2,\dots,\lambda_n$. Can we express the eigenvalues of $I-A$ using eigenvalues of $A$? I can't find properties of eigenvalue related to this problem.Suppose we could dia

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