Recent content by Steve Turchin

Homework Statement ##\int_{-\infty} ^{\infty} dx \int_{-\infty} ^{\infty} dy \ \ e^{-3x^2+2xy-3y^2} ## Homework Equations ## \int_{-\infty} ^{\infty} dx \int_{-\infty} ^{\infty} dy \ \ e^{-x^2-y^2} = \pi ## The Attempt at a Solution ##3x^2-2xy+3y^2 = (x,y) \left( \begin{array}{ccc} 3 & -1 \\...

Is the basis vector ##(i,0,1)## in the space ##V=##Span##((i,0,1))## with a standard inner product,over ##\mathbb{C}^3## orthogonal to itself? ##<(i,0,1),(i,0,1)> = i \cdot i + 0 \cdot 0 + 1 \cdot 1 = -1 + 1 = 0 ## The inner product (namely dot product) of this vector with itself is equal to...

Thank you for the reply. It's a sum of ## n ## terms. Any tips on how to calculate an exact form of the sum: ## \sum_{k=1}^{n} ke^{\frac{k}{n}} ## ?

Homework Statement ## lim_{n \rightarrow \infty}{\frac{1}{n^2} \sum_{k=1}^{n} ke^{\frac{k}{n}}} ## Homework EquationsThe Attempt at a Solution ## lim_{n \rightarrow \infty}{\frac{1}{n^2} \sum_{k=1}^{n} ke^{\frac{k}{n}}} \\ = lim_{n \rightarrow \infty}{\frac{1}{n^2}...

Thanks a lot vela! ## p(2)=16a_4+8a_3+4a_2+2a_1+a_0=0 ## ## p(1)-p(i)=0=a_4+a_3+a_2+a_1+a_0-(a_4-ia_3-a_2+ia_1+a_0) ## ## (1+i)a_3+2a_2+(1-i)a_1=0 \ \ \Rightarrow \ a_3=\frac{(i-1)a_1-2a_2}{1+i} ## Solving the first equation: ##16a_4+8(\frac{i-1}{1+i}a_1-\frac{2}{1+i}a_2)+4a_2+2a_1+a_0 = 0...

Thank you so much for the thorough reply and sorry for posting this on the wrong forum. Solving ## p(1)=p(i) ## ## (1-2)(c_3+c_2+c_1+c_0)=(i-2)(c_3 i^3+c_2 i^2+c_1 i+c_0) ## I get: ## -(c_3+c_2+c_1+c_0)+i(c_0-c_2+2c_3-2c_1)+c_3-c_1+2c_2-2c_0 ## ##c_0-c_2+2c_3-2c_1=0 ## ##2c_3+3c_2-c_0=0 ##...

Homework Statement Find a basis for the following vector space: ## V = \{ p \in \mathbb C_{\leq4} ^{[z]} | \ p(1)=p(i) ## and ## p(2)=0 \} ## (Where ## \mathbb C_{\leq4} ^{[z]} ## denotes the polynomials of degree at most 4) Homework Equations N/A The Attempt at a Solution I tried to find...

Homework Statement Prove that if ## |x-x_0|<\min (\frac {\epsilon}{2(|y_0|+1)},1)## and ##|y-y_0|<\frac{\epsilon}{2(|x_0|+1)} ## then ## |xy-x_0y_0|<\epsilon ## Homework Equations N/A The Attempt at a Solution From the first inequality I can see that: ## |x-x_0|<\frac...

I think a possible answer would be the limit of ## f \circ g ## as ## g(x)=x^3-x ## . ## \lim_{x\to 0} x^3-x = 0 ## I believe that I need to know whether or not the function is continuous at ## x=0 ##. Thanks for the replies. Here's what I tried next, using the chain limit rule which andrewkirk...

Homework Statement If ## lim_{x\to 0^+} f(x) = A ## and ## lim_{x\to 0^-} f(x) = B ##, find: (a) ## lim_{x\to 0^+} f(x^3-x) ## (b) ## lim_{x\to 0^-} f(x^3-x) ## Homework Equations None. Only the problem statement. The Attempt at a Solution (a) ## lim_{x\to 0^+} f(x^3-x) = lim_{x\to 0^+}...

Solve the following inequality. Represent your answer graphically: ## |z-1| + |z-5| < 4 ## Homework Equations ## z = a + bi \\ |x+y| \leq |x| + |y| ## Triangle inequality The Attempt at a Solution ## |z-1| + |z-5| < 4 \\ \\ x = z-1 \ \ , \ \ y = z-5 \\ \\ |z-1+z-5| \leq |z-1| + |z-5| \\...