Deriving inverse tans from complex product?

In summary, to derive 4\,\arctan \left( 1/5 \right) -\arctan \left( {\frac {1}{239}} \right) from the complex product of \left( 1+i \right) \left( 5-i \right) ^{4}, you need to expand the product as (5-i)^{2^2}\cdot(1+i) and then multiply it out, resulting in 956-4i. The argument of this complex number is equal to -\arctan \left( {\frac {1}{239}} \right), which is the desired result. This method is a derivation of the famous Machin's formula for
  • #1
rsnd
26
0
I am supposed to derive [itex]4\,\arctan \left( 1/5 \right) -\arctan \left( {\frac {1}{239}}
\right) [/itex] from the complex product of [itex]\left( 1+i \right) \left( 5-i \right) ^{4}[/itex] I do see how the argument of product of the complex expression is equal to pi/4 - 4 arctan(1/5) but I am totally lost. SO how am I supposed to approach this? Thanks heaps in advance.
 
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  • #2
Well You have to equate to long way and the short way :D

You already established that the argument to the complex product is [itex]\frac{\pi}{4} -4\arctan \frac{1}{5}[/itex]

What you had to do next is have the pleasure of EXPANDING YOUR PRODUCT. Binomial theorem is just too much of a hassle this time. Write the product as
[tex](5-i)^{2^2}\cdot(1+i)[/tex]. So just square 5-i, 2 times, and multiply that by 1+i.

FUN FUN, I did it for you :P You get : 956-4i . The argument of this is arctan (-4/956)= - arctan (1/239) . Isn't that interesting?
 
  • #3
And here I thought I'd never have to multiply the hard way after learning the polar form. Thanks heaps again =)
 
  • #4

1. What is the purpose of deriving inverse tans from complex product?

The purpose of deriving inverse tans from complex product is to solve for the inverse tangent of a complex number, which is useful in various mathematical and scientific calculations.

2. How do you derive inverse tans from complex product?

To derive inverse tans from complex product, you can use the formula atan(z) = 1/2i * ln((i+z)/(i-z)), where z is a complex number.

3. What are the properties of inverse tans from complex product?

The properties of inverse tans from complex product include the fact that it is a multi-valued function, has branch cuts at ±i, and is an odd function. It also follows the same rules of inverse trigonometric functions, such as the double angle formula and the sum and difference formulas.

4. How do you use inverse tans from complex product in real-life applications?

Inverse tans from complex product are used in various fields such as engineering, physics, and computer science. They are particularly useful in solving problems involving complex numbers, such as in electrical circuits, signal processing, and control systems.

5. Are there any limitations to using inverse tans from complex product?

One limitation of using inverse tans from complex product is that it can only be applied to complex numbers, and not real numbers. Additionally, the multi-valued nature of the function can lead to some ambiguity in certain calculations and may require additional steps to obtain the desired result.

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