# Tagged Questions

2answers
50 views

### Different Definitions Of The Sine Function

I was hoping someone could give me a flow chart or high-level map connecting all of the definitions of the sine function, with some of the reasons why we care next to each. I've tried this but I'm not ...
1answer
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5answers
263 views

### When are we (not) allowed to replace $x$ by $ix$?

It seems to be quite a common manipulation to replace $x$ by $ix$. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...
3answers
206 views

### Could we show $1-(x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dots)^2=(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}- \dots)^2$ if we didn't know about Taylor Expansion?

Suppose that humanity haven't discovered Taylor Series Expansion of trigonometric functions or of any function that would help us on this. Which means we are not allowed to replace the given infinite ...
1answer
119 views

### Power series for $\cos(n\theta)$ in terms of $\sin^{2i}(\theta/2)$?

Does anybody know an expression for the weights in $$\cos(n\theta) = \sum_{i=0}^n c_i \sin^{2i}(\theta/2)$$ I checked the standard sources (Abramowitz & Stegun, Gradshteyn & Rhyzik) and ...
1answer
136 views

### Maclaurin expansion of $\sqrt{\cos 2x}$ and $\tan^2 x$ up to degree 4

Find the Maclaurin expansion $\sqrt{\cos(2x)}$ and $\tan^2x$ up to degree $4$. I tried differentiation but it gives me something really horrible.
2answers
334 views

### How to do a very long division: continued fraction for tan

I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in ...
1answer
153 views

### Evaluate $\ \int_{- \infty}^{\infty} \frac{x}{( x^2 + 4x + 13 )^2}\,dx . \$ using contour integration and the calculus of residues

How do I evaluate $\ \int_{- \infty}^{\infty} \frac{x}{( x^2 + 4x + 13 )^2}\,dx . \$ using contour integration and the calculus of residues? I have many more problems of this kind need to be done. ...
2answers
151 views

### Use Residue Theorem to evaluate $\ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \$?

How do I use Residue Theorem to evaluate $\ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \$ where $C_3(0)$ is the circle of radius 3 centered at the origin, oriented in the counter- clockwise ...
1answer
329 views

### Cauchy product and the exponential function

Simplify the following series using the Cauchy product $$\sum\limits_{k=1}^\infty\frac{1}{k!}\cdot\sum\limits_{j=1}^\infty\frac{1}{j!}$$ ...
2answers
183 views

### How to transform the factored form of $\sin(x)$?

We know $\sin(x)=0$ has solutions $0,\pm\pi,\pm2\pi,\pm3\pi,\dots$. So $\sin(x)$, if interpreted as a polynomial, could be written as: $a_0x^0+a_1x^1+a_2x^2+\cdots$ and we know this polynomial too: ...
2answers
304 views

### Identify this power series / solve this trig equation

I was asked to find a solution to $$\frac{\sin^2(nx)}{n^2\sin^2(x)}=2^{-1/2}$$ where $n$ is a fixed integer greater than 1. Numerically, there's a solution just above 1/n so I decided to find this ...