LambertW(x,y) Umstellungsbeispiele


Grundregeln:

§A: x * x = x hoch 2 = x² = x^2 = pow(x,2) = e^(log(x)*2)=exp(log(x)*2)

§B: f(x) = e^x = exp(x) ;

Umkehrfunktion: f -1(x)=log(x)

§C: f(x) = x * e^x = x*ex;

Umkehrfunktion: f -1(x)=LambertW(-½±½,±x)

§D: f(x) = x^n * e^x = xn * ex;

Umkehrfunktion: f -1(x)=LambertW(-½±½,±(±x)^(1/n)/n)*n; n>1

§E: f(x) = x^(1/n) * e^x = x(1/n) * ex;

Umkehrfunktion: f -1(x)=LambertW(-½±½,±(±x)^n*n)/n; n>1



§1 1.5^x = 2x-1

(3/2)^x=e^(log(3/2)*x)=2x-1

|*log(3/2)/2

log(3/2)/2*e^(log(3/2)*x)=log(3/2)*x-log(3/2)/2

| /(-e^(log(3/2)*x))

-log(3/2)/2 = [log(3/2)/2-log(3/2)*x]*e^(-log(3/2)*x)

|Substitution1: u=-log(3/2)*x

-log(3/2)/2 = [log(3/2)/2+u]*e^u = log(3/2)/2*e^u + u *e^u

|*e^[log(3/2)/2]

-log(3/2)/2*e^[log(3/2)/2] = [log(3/2)/2+u]*e^(log(3/2)/2+u)

|Subst2: v=log(3/2)/2+u

-sqrt(3/2)*log(3/2)/2= v*e^v

| Umkehrfunktion §C

v=LambertW(-sqrt(3/2)*log(3/2)/2)

| Rücksubst2 mit v

log(3/2)/2+u=LambertW(-sqrt(3/2)*log(3/2)/2)

|- log(3/2)/2

u=W(-sqrt(3/2)*log(3/2)/2)-log(3/2)/2

|RückSubst.1 mit u

-log(3/2)*x=LambertW(-sqrt(3/2)*log(3/2)/2)-log(3/2)/2

|/(-log(3/2))

x1=[log(3/2)/2-LambertW(0,-sqrt(3/2)*log(3/2)/2)]/log(3/2)

x1=1.3721575818167754273229258411755523362909808531253359...

x2=[log(3/2)/2-LambertW(-1,-sqrt(3/2)*log(3/2)/2)]/log(3/2)

x2=5.8420922416969373830452071896430762958177061238753367...


allgemein: a^x=b*x+c
x=-c/b-LambertW(n , -a^(-c/b)*log(a)/b)/log(a) ; n=-2...2; b*log(a)<>0


§2 a^x=x^n -> x=-(n*LambertW(-½±½,-(log(a))/n))/(log(a))


Beispiel: a=11/10 und n=2

x1=-(2*LambertW( 0,-(log(11/10))/2))/(log(11/10))

=1.0513800237472769373675721833547129751259449698675902...

x2=-(2*LambertW(-1,-(log(11/10))/2))/(log(11/10))

=95.716830168405222740003823032089546239002719059429431...

wegen geradzahligen ganzen Potenz (n mod 2 =0), auch noch +:

x3=-(2*LambertW( 0,+(log(11/10))/2))/(log(11/10))

=-0.95548727594562198165088501460980486941793244835341...

jetzt noch komplexe Anteile:

LambertW(-1,0.0476550899021624300219760616403825461103026826543220)

=-4.864454383150863212294516115508091-3.805445106738860023838468740911048i

x4=-(2*LambertW(-1,(log(11/10))/2))/(log(11/10))=

=-(2*(-4.864454383150863212294516115508091-3.805445106738860023838468740911048i))/(log(11/10))

=102.0762817390074925895879903437157 +79.85390678207872898044462313213559 i

Kontrolle:

(11/10)^(102.0762817390074925895879903437+79.85390678207872898044462313213 i)-(102.0762817390074925895879903437157+79.85390678207872898044462313213559 i)²

=0


§3 x^x=y


x^x=y

| x. Wurzel

x=y^(1/x)

| /x

1=1/x * y^(1/x)

| §A und Subst.1 u=1/x

1= u * e^(log(y) * u)

| *log(y)

log(y)=log(y) * u * e^(log(y) * u)

| subst.2: v= log(y) * u

log(y)= v * e^v

| §C

v=LambertW(log(y))

| Rücksubst. 2 mit v → u

log(y) * u = LambertW(log(y))

| Rücksubst 1 mit u → x

log(y) * 1/x = LambertW(log(y))

| *x/LambertW

x = log(y) / LambertW(-½ ± ½,log(y))


§4 log(2x-1)*(1-2x)=2x


0=2x+log(2x-1)*(2x-1)

| -1

-1=(2x-1)+log(2x-1)*(2x-1)

| Subst1: u=2x-1 und Exponentialfunktion

exp((-1-u)/u)=u

|*(-1/u)

e^(-1/u-1)*(-1/u)=-1

| *e^1

e^(-1/u)*(-1/u)= -e

| Subst2: v=-1/u und §C

v=LambertW(-e)

| Rücksubst. 2 mit v → u

u=-1/LambertW(-e)

| Rücksubst 1 mit u → x

2x-1=-1/LambertW(-e)

| +1 dann /2

x=(1-1/LambertW( - ½± ½,-e))/2


X1= 0.44111161967499482446076447024089... +0.26660525097138465444099823482124... i

x2= 0.44111161967499482446076447024089... -0.26660525097138465444099823482124... i


§5   e^(a*x) = b*x + c

e^(a*x)=b*x+c

|/e^(a*x)

1=(b*x+c)*e^(-a*x)

| *(-a/b)

-a/b=(-a*x-a*c/b)*e^(-a*x)

| /e^(a*c/b)

-a/[b*e^(a*c/b)]=(-a*x-a*c/b)*e^(-a*x-a*c/b)

|Substitution: u=-a*x-a*c/b

-a/[b*e^(a*c/b)]=u*e^u

|Umkehrfunktion §C

u=LambertW(-½ ± ½ ,-a/[b*e^(a*c/b)])

| RückSubst. mit u

-a*x-a*c/b=LambertW(-½ ± ½ ,-a/[b*e^(a*c/b)])

| +a*c/b

-a*x=W(...)+a*c/b

| /(-a)

x=-LambertW(-½ ± ½ , -a/[b*e^(a*c/b)]) /a - c/b



§6   e^(a*x+p)*x^h = b

x=h/a * LambertW(n, a/h * (-1)^(2*N/h) * (b/e^p)^(1/h)) mit n=-2,-1,0,1; h=1,2,3,4,...; N=0,1,2,...,(h-1)



§7   x^a = b*log(x)

x=e^[-LambertW(n, -a/b)/a] mit n=-1,0,1 je nach a >0 oder kleiner oder a>b/e oder kleiner...



§8   (x+a)* b^x = c

(x+a)* b^x = c                 |*b^a
(x+a)* b^(x+a) = c *b^a        | Subst.: u=x+a; b^u=e^(log(b)*u)
u * e^(log(b)*u) = c *b^a      | *log(b) und subst.: z=log(b)*u
z * e^z = c *b^a*log(b)        | §C Umkehrfunktion
z=LambertW(n,c *b^a*log(b))    |Rücksubst.:
u=LambertW(n,c *b^a*log(b))/log(b)  | Rück2
x=LambertW(n,c *b^a*log(b))/log(b)-a ; n=-2...1
mit a=2, b=3, c=4
 n | x 4 Lösungen:
-2 | -0.85000206103882985793215861..-10.1116813238567445075536415439845419914... i
-1 | -0.19698146687223427792235753..-4.62758241481090444814100263917140473413... i
 0 |  0.44723345968841609331415512...
 1 | -0.19698146687223427792235753..+4.62758241481090444814100263917140473413... i



§9   ln(a*x + b)+c*x/(a*x+b)=0

ln(a*x + b)+c*x/(a*x+b)=0               |*(a*x+b)
(a*x+b)*ln(a*x + b)=-c*x                | Subst.: u=ln(a*x+b); e^u=a*x+b; x=(e^u-b)/a
e^u * u =-c*(e^u-b)/a=c*b/a-e^u*c/a     |+e^u*c/a
e^u *(u+c/a) = c*b/a                    | *e^(c/a)
(u+c/a) * e^(u+c/a) = c*b/a*e^(c/a)     | Subst2: z=u+c/a + Umkehrfunktion
z = LambertW(c*b/a*e^(c/a))             | Rücksubst.2 und -c/a
u = LambertW(e^(c/a)*c*b/a)-c/a         | Rücksubst. u=ln(a*x+b)
ln(a*x+b)= LambertW(e^(c/a)*c*b/a)-c/a  | e^...
a*x+b = e^[LambertW(e^(c/a)*c*b/a)-c/a] | -b dann /a
x={e^[LambertW(e^(c/a)*c*b/a)-c/a]-b}/a
Hinweis: hier gibt es meist keine mehrfach-komplexen Nullstellen, also n=0, was man auch weglassen kann
(also immer Probe)


Liste von Rechnern mit der Lambert W Funktion (List of calculators with Lambert W function)
Rechner mit LambertW/Polylog Nachkommastellen W(x) W(n,y) komplex kostenlos online
Umkehrfunktionen Rechner 32...34 v v v v
WolframAlpha 6...3600 v v v v
HP‐41Z 3…9 v   v  
HP 49/50   v      
HP WP 34S   v      
Hp 5971 ?          

zurück zum Umkehrfunktionen Rechner: http://www.lamprechts.de/gerd/php/RechnerMitUmkehrfunktion.php