Helu Euler (e)

Number e (a i ʻole, e like me ia i kapa ʻia, ka helu Euler) ke kumu o ka logaritma kūlohelohe; he helu makemakika, he helu kuhi hewa.

e = 2.718281828459 …

Nā ala e hoʻoholo ai i ka helu e (ka hoʻohālikelike):

1. Ma o ka palena:

ʻO ka palena kupaianaha lua:

ʻO kahi koho ʻē aʻe (e hahai ana mai ke ʻano De Moivre-Stirling):

2. E like me ka huina pūʻulu:

waiwai helu e

1. Ka palena pānaʻi e

2. Nā huaʻōlelo

ʻO ka derivative o ka hana exponential ka hana exponential:

(e ^x)′ = a^x

ʻO ka derivative o ka hana logarithmic kūlohelohe ka hana inverse:

(log_e x)′ = (ln x)′ = 1/x

3. Huina

ʻO ka hoʻohui pau ʻole o ka hana exponential e ^x he hana exponential e ^x.

∫ a^x dx = e^x+c

ʻO ka hoʻohui pau ʻole o ka logarithmic function log_e x:

∫ log_e x dx = ∫ lnx dx = x ln x – x +c

Huipuia maopopo o 1 i e Ua like ka hana inverse 1/x me 1:

Logarithms me ke kumu e

Logarithm kūlohelohe o kahi helu x wehewehe ʻia ʻo ia ka logarithm kumu x me ke kumu e:

ln x = kūloko_e x

Hana Hoʻonui

He hana exponential kēia, i wehewehe ʻia penei:

f (x) = exp(x) = e^x

Euler formula

Helu paʻakikī e ^iθ like:

e^iθ = cos (θ) + i hewa (θ)

kahi i ʻo ia ka ʻāpana noʻonoʻo (ke kumu huinahalike o -1), a θ he helu maoli.