8.4.2 Greatest hits from previous weeks

8.4.2.1 Multivariable Taylor expansion does not work as expected
8.4.2.2 Extract real part of a complex expression better than Re does
8.4.2.3 1 Plot, 2 Scale/Axis
8.4.2.4 What does # mean in Mathematica?
8.4.2.5 All that doesn't glitter isn't gold
8.4.2.6 How can I create an empty JSON object?
8.4.2.7 How to perform a multi-peak fitting?
8.4.2.1 Multivariable Taylor expansion does not work as expected

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=15023

The basic multivariable Taylor expansion formula around a point is as follows: 
 
$$ 
f(\mathbf r + \mathbf a) = f(\mathbf r) + (\mathbf a \cdot \nabla )f(\mathbf r) + \frac{1}{2!}(\mathbf a \cdot \nabla)... 
[calculus-and-analysis] [series-expansion]
 
asked by matheorem https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=4742 Score of 70
answered by Jens https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=245 Score of 78
8.4.2.2 Extract real part of a complex expression better than Re does

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=3306

I have a complex expression with real positive variables only. 
 
Mathematica Input Style: 
 
... 
[symbolic] [trigonometry]
 
asked by max https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=796 Score of 36
answered by nixeagle https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=814 Score of 56
8.4.2.3 1 Plot, 2 Scale/Axis

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=627

I would like to plot those two datasets on top of each other. But they have very different range on the $y$ axis. How can I have two different axis? 
 
I found the following on the help menu but quite ... 
[plotting] [customization]
 
asked by 500 https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=172 Score of 156
answered by ArgentoSapiens https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=92 Score of 157
8.4.2.4 What does # mean in Mathematica?

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=19035

I asked Mathematica to compute the following 
 
 Solve[c (1-x)^2-x^(1/4) == 0, x] 
 
 
and it returned this: 
x = Root[#1^8 c^4 - 8 #1^7 c^4 + 28 #1^6 c^4 - 56 #1^5 c^4 ... 
[syntax] [core-language]
 
asked by ronanymous https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=5730 Score of 43
answered by Dr. belisarius https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=193 Score of 74
8.4.2.5 All that doesn't glitter isn't gold

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=249714

I would like to color some objects in metallic shades like gold and silver. Various web sources suggest that the gold is a variant of yellow with RGB values 83.14% red, 68.63% green, and 21.57% blue. ... 
[graphics] [color]
 
asked by bill s https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=1783 Score of 22
answered by Tim Laska https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=61809 Score of 45
8.4.2.6 How can I create an empty JSON object?

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=50727

For creating non-empty object, I use following syntax 
 
 ExportString[{"key" -> "value"}, "JSON"] 
(*{"key" : "value"}*) 
 
 
But how can I create an empty object? ... 
[export] [conversion] [json-format]
 
asked by molekyla777 https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=11981 Score of 10
answered by jkuczm https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=14303 Score of 11
8.4.2.7 How to perform a multi-peak fitting?

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=26336

I am wondering how to implement the multi-peak detecting and fitting in Mathematica. Following is an example of fitting the data using three peaks (such that the data ~ peak1 + peak2 + peak3). 
The ... 
[fitting] [data] [peak-detection]
 
asked by Everett You https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=1389 Score of 56
answered by Silvia https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=17 Score of 55