Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 100584 by Dwaipayan Shikari last updated on 27/Jun/20

∫i^i^(i......∞)  dx

$$\int{i}^{{i}^{{i}......\infty} } {dx} \\ $$

Answered by MJS last updated on 27/Jun/20

∫zdx=z∫dx=zx+C  in this case: ∫i^i^(i...∞)  dx=x×i^i^(i...∞)  +C

$$\int{zdx}={z}\int{dx}={zx}+{C} \\ $$$$\mathrm{in}\:\mathrm{this}\:\mathrm{case}:\:\int\mathrm{i}^{\mathrm{i}^{\mathrm{i}...\infty} } {dx}={x}×\mathrm{i}^{\mathrm{i}^{\mathrm{i}...\infty} } +{C} \\ $$

Commented by Dwaipayan Shikari last updated on 27/Jun/20

Sir can you calculate the value of i^i^i^(i...∞)   ?

$${Sir}\:{can}\:{you}\:{calculate}\:{the}\:{value}\:{of}\:{i}^{{i}^{{i}^{{i}...\infty} } } ? \\ $$

Commented by Dwaipayan Shikari last updated on 27/Jun/20

My main question is to find out the value  of  i^i^(i..∞)

$${My}\:{main}\:{question}\:{is}\:{to}\:{find}\:{out}\:{the}\:{value}\:\:{of}\:\:{i}^{{i}^{{i}..\infty} } \\ $$

Commented by MJS last updated on 27/Jun/20

usually  x^x^(x...)  =y  ⇒  x^y =y  y ln x =ln y  −((ln y)/y)=−ln x  y=e^(−t)   e^t t=−ln x  we need the Lambert W function, I don′t  know if there are solutions for x∉R

$$\mathrm{usually} \\ $$$${x}^{{x}^{{x}...} } ={y} \\ $$$$\Rightarrow \\ $$$${x}^{{y}} ={y} \\ $$$${y}\:\mathrm{ln}\:{x}\:=\mathrm{ln}\:{y} \\ $$$$−\frac{\mathrm{ln}\:{y}}{{y}}=−\mathrm{ln}\:{x} \\ $$$${y}=\mathrm{e}^{−{t}} \\ $$$$\mathrm{e}^{{t}} {t}=−\mathrm{ln}\:{x} \\ $$$$\mathrm{we}\:\mathrm{need}\:\mathrm{the}\:\mathrm{Lambert}\:\mathrm{W}\:\mathrm{function},\:\mathrm{I}\:\mathrm{don}'\mathrm{t} \\ $$$$\mathrm{know}\:\mathrm{if}\:\mathrm{there}\:\mathrm{are}\:\mathrm{solutions}\:\mathrm{for}\:{x}\notin\mathbb{R} \\ $$

Commented by MJS last updated on 27/Jun/20

approximately I get  y≈.438283+.360592i  using a good calculator starting with  i  and then typing “i^(last answer) ”

$$\mathrm{approximately}\:\mathrm{I}\:\mathrm{get} \\ $$$${y}\approx.\mathrm{438283}+.\mathrm{360592i} \\ $$$$\mathrm{using}\:\mathrm{a}\:\mathrm{good}\:\mathrm{calculator}\:\mathrm{starting}\:\mathrm{with} \\ $$$$\mathrm{i} \\ $$$$\mathrm{and}\:\mathrm{then}\:\mathrm{typing}\:``\mathrm{i}^{\mathrm{last}\:\mathrm{answer}} '' \\ $$

Commented by Dwaipayan Shikari last updated on 27/Jun/20

Thanking you for interacting

$${Thanking}\:{you}\:{for}\:{interacting} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com