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Dec
16
comment Is the derivative of $\{x\}$ on $(0,1)$ always equal to $1$?
Yes, sorry - question updated. Thanks for spotting this.
Dec
3
comment Summation of logarithms
You have $$S=\sum_{k=1}^n\log(a-x_k)=\log\prod_{k=1}^n(a-x_k).$$ So $$\prod_{k=1}^n(a-x_k)-e^S=0.$$ So you need to be able to solve equations of the form: $$\prod_{k=1}^n(a-x_k)-c=0,$$ where $c$ is some constant.
Nov
20
comment Finding the roots of $x^2+(3+5i)x+(7+11i)=0$
Re polar exponential form - see my answer below for the answer.
Nov
19
comment Most ambiguous and inconsistent phrases and notations in maths
Since I used $\log$ instead of $\text{Log}$ and I did not specify a branch then to me this is clearly a mistake.
Nov
19
comment Most ambiguous and inconsistent phrases and notations in maths
@Mariano Suárez-Alvarez The notation $\text{Log}$ is also sometimes reserved for the principal branch, often when $\log$ for the (general) multi-valued function is used. In whichever context, I've seen this cause confusion in obtaining results.
Nov
15
comment What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)
In fact: $e^{ix}=\cos(x)+i\sin(x)$.
Nov
13
comment What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)
Good point. I think at the time I "understood" what numbers were, but these new numbers (complex numbers) were so strange, and were presented with the added "imaginary" verbiage that they were mesmerizing to me! Of course later I realised that complex numbers are not complex at all and that there is absolutely nothing imaginary about the idea of $i$. Complex numbers are a system of numbers which obey a certain set of rules in a consistent way. But I'm pleased to have been captivated :-)
Nov
11
comment The Integral of Multiple Tangent Functions
Try increasing the working precision: Integrate[..., WorkingPrecision->100].
Nov
11
comment Is the derivative of $\{x\}$ on $(0,1)$ always equal to $1$?
So it's down to the method used to compute the derivative. Need to be judicious with tools.
Nov
10
comment Is the derivative of $\{x\}$ on $(0,1)$ always equal to $1$?
Mathematica is great, but it does have bugs. I've found at least 5 so far in my life time of using it. From infinite summations, to sum convergence conditions, to this !! I've filed another bug report with Mathematica ;-)
Oct
24
comment Nature of the range of $e^x$
I've updated the question. Hope that helps.
Oct
21
comment calculus first impressions
It should be a bit more than just algebraic calculations. Did you not see a limit $\lim$ in the maths somewhere? Do you really understand the limiting process? I'd try some examples and don't jump the gun too soon. Take advantage of the "spare" time you have if you find it easy to understand :-) Or read on further so you will then understand the later lectures!
Oct
20
comment Specific form of differential equation
Does this type of differential equation have a name?
Oct
20
comment Specific form of differential equation
I reworded my question. Hopefully its clearer now. Sorry if its still unclear.
Oct
13
comment How do I choose between $\lim_{x\to a} \frac {f(x) - f(a)}{x-a}\ $ and $\lim_{x\to a} \frac{f(a+h)-f(a)}{h}$?
To avoid the symbolic "overkill" you could always write the short hand notation for the binomial expansion, e.g. $$(x+h)^4=\sum_{k=0}^4{4\choose k}x^{4-k}h^k,$$ especially for large powers.
Oct
12
comment Find the limit of a Riemann Sum
$ f (X)=(1-X)(1+X) $.
Oct
7
comment Calculating a complex definite improper integral: $I= \int_{0}^\infty x^{it}\,\mathrm{e}^{-ax}\, dx$
For a general treatment of these types of integrals, you may be interested in learning about the Bilateral Laplace Transform: en.wikipedia.org/wiki/…
Oct
3
comment how to calculuate $\int_0^ \pi \sqrt{1+x^2 \sin^2x}dx$
For what it's worth the integrand is symmetric, so we may write $$I = \frac{1}{2}\int_{-\pi}^\pi\sqrt{1+(x\sin x)^2}dx.$$
Oct
2
comment Change of variable in complex integral
Your substitution should be $z=Re^{i\theta}$, so that $dz=Rie^{i\theta}d\theta$, and $0\leq \theta< \pi/4$. We make this substitution because the substitution describes the path $\Gamma$ which you give.
Oct
2
comment What is the relation between two integrals?
On the real line you would have $I_1\leq I_2$. But from your language it's unclear whether you're talking about integration over $\mathbb{R}$ or $\mathbb{C}$.