Any engineers (aerospace) here? (solution to theta beta mach relation)

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dmw16

Diamond Member
Nov 12, 2000
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I have a rather challenging problem. I am writing a program that will model blunt body reentry into the earth's atmosphere, but I need to be able to pull out values of beta from the theta beta mach relation. Is there anyone here who possibly knows a numerical solution or is good w/ mathematic and can solve this function for me for beta....

tan(theta) = 2*cot(beta)*[(M^2 * (sin(beta))^2 - 1) / (M^2 * (gamma + cos(2*beta)) + 2)]

thanks.
-doug
 
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RaynorWolfcastle

Diamond Member
Feb 8, 2001
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how much do we expect beta to vary here? You may be able to get away with a power series expansion of all the trig terms if the variance isn't too big.

<-- 2nd year EE student, has never heard of this relation before
 
D

Descartes

Lifer
Oct 10, 1999
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Originally posted by: DorkBoy
I'm a Nuclear Eng. but I don't remember sh*t
Click to expand...

That's comforting. I'm a doctor, but I don't remember anything either. I'm offering discounts for patients I get from AT.
 
J

jaydee

Diamond Member
May 6, 2000
4,500
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81
Originally posted by: Descartes
Originally posted by: DorkBoy
I'm a Nuclear Eng. but I don't remember sh*t
Click to expand...

That's comforting. I'm a doctor, but I don't remember anything either. I'm offering discounts for patients I get from AT.
Click to expand...

SWEET! Someone post this in Hot Deals, and mods can we get a sticky?!
 
R

RaynorWolfcastle

Diamond Member
Feb 8, 2001
8,968
16
81
Using Maple's solve command:

beta = arctan(1/6*(6/M^2/(tan(theta)^2+ 1)*(- 8- 36*tan(theta)^4*M^4- 24*tan(theta)^4- 60*tan(theta)^4*M^4*gamma- 24*tan(theta)^4*M^2*gamma- 24*tan(theta)^4*M^4*gamma^2+ 8*M^6- 9*tan(theta)^4*M^6*gamma^3+ 6*tan(theta)^4*M^6+ 15*tan(theta)^2*M^6- 12*tan(theta)^6*M^2- 6*tan(theta)^6*M^4- tan(theta)^6*M^6- 12*tan(theta)^6*M^4*gamma- 12*tan(theta)^6*M^2*gamma- 6*tan(theta)^6*M^4*gamma^2- 9*tan(theta)^2*M^6*gamma^2+ 6*tan(theta)^2*M^6*gamma- 3*tan(theta)^6*M^6*gamma^2- tan(theta)^6*M^6*gamma^3- 3*tan(theta)^6*M^6*gamma- 12*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^6+ 3*tan(theta)^4*M^6*gamma+ 3*3^(1/2)*tan(theta)*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 3*3^(1/2)*tan(theta)^3*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 24*M^2- 24*M^4- 48*tan(theta)^2*M^4*gamma- 12*tan(theta)^2*M^2*gamma- 18*tan(theta)^2*M^4*gamma^2- 54*tan(theta)^2*M^4+ 36*tan(theta)^2*M^2- 24*tan(theta)^2)^(1/3)+ 6*(2*tan(theta)^4*M^4*gamma+ 4*tan(theta)^4*M^2*gamma+ tan(theta)^4*M^4*gamma^2+ 4*tan(theta)^4*M^2+ tan(theta)^4*M^4+ 4*tan(theta)^4- 8*M^2+ 4*M^4+ 4- 3*tan(theta)^2*M^4*gamma^2+ 2*tan(theta)^2*M^4*gamma+ 4*tan(theta)^2*M^2*gamma+ 5*tan(theta)^2*M^4- 4*tan(theta)^2*M^2+ 8*tan(theta)^2)/M^2/(tan(theta)^2+ 1)/(- 8- 36*tan(theta)^4*M^4- 24*tan(theta)^4- 60*tan(theta)^4*M^4*gamma- 24*tan(theta)^4*M^2*gamma- 24*tan(theta)^4*M^4*gamma^2+ 8*M^6- 9*tan(theta)^4*M^6*gamma^3+ 6*tan(theta)^4*M^6+ 15*tan(theta)^2*M^6- 12*tan(theta)^6*M^2- 6*tan(theta)^6*M^4- tan(theta)^6*M^6- 12*tan(theta)^6*M^4*gamma- 12*tan(theta)^6*M^2*gamma- 6*tan(theta)^6*M^4*gamma^2- 9*tan(theta)^2*M^6*gamma^2+ 6*tan(theta)^2*M^6*gamma- 3*tan(theta)^6*M^6*gamma^2- tan(theta)^6*M^6*gamma^3- 3*tan(theta)^6*M^6*gamma- 12*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^6+ 3*tan(theta)^4*M^6*gamma+ 3*3^(1/2)*tan(theta)*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 3*3^(1/2)*tan(theta)^3*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 24*M^2- 24*M^4- 48*tan(theta)^2*M^4*gamma- 12*tan(theta)^2*M^2*gamma- 18*tan(theta)^2*M^4*gamma^2- 54*tan(theta)^2*M^4+ 36*tan(theta)^2*M^2- 24*tan(theta)^2)^(1/3)+ 12/M^2*(tan(theta)^2*M^2*gamma+ 2*tan(theta)^2+ 2+ tan(theta)^2*M^2+ M^2)/(tan(theta)^2+ 1))^(1/2),1/12*tan(theta)*(6/M^2/(tan(theta)^2+ 1)*(- 8- 36*tan(theta)^4*M^4- 24*tan(theta)^4- 60*tan(theta)^4*M^4*gamma- 24*tan(theta)^4*M^2*gamma- 24*tan(theta)^4*M^4*gamma^2+ 8*M^6- 9*tan(theta)^4*M^6*gamma^3+ 6*tan(theta)^4*M^6+ 15*tan(theta)^2*M^6- 12*tan(theta)^6*M^2- 6*tan(theta)^6*M^4- tan(theta)^6*M^6- 12*tan(theta)^6*M^4*gamma- 12*tan(theta)^6*M^2*gamma- 6*tan(theta)^6*M^4*gamma^2- 9*tan(theta)^2*M^6*gamma^2+ 6*tan(theta)^2*M^6*gamma- 3*tan(theta)^6*M^6*gamma^2- tan(theta)^6*M^6*gamma^3- 3*tan(theta)^6*M^6*gamma- 12*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^6+ 3*tan(theta)^4*M^6*gamma+ 3*3^(1/2)*tan(theta)*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 3*3^(1/2)*tan(theta)^3*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 24*M^2- 24*M^4- 48*tan(theta)^2*M^4*gamma- 12*tan(theta)^2*M^2*gamma- 18*tan(theta)^2*M^4*gamma^2- 54*tan(theta)^2*M^4+ 36*tan(theta)^2*M^2- 24*tan(theta)^2)^(1/3)+ 6*(2*tan(theta)^4*M^4*gamma+ 4*tan(theta)^4*M^2*gamma+ tan(theta)^4*M^4*gamma^2+ 4*tan(theta)^4*M^2+ tan(theta)^4*M^4+ 4*tan(theta)^4- 8*M^2+ 4*M^4+ 4- 3*tan(theta)^2*M^4*gamma^2+ 2*tan(theta)^2*M^4*gamma+ 4*tan(theta)^2*M^2*gamma+ 5*tan(theta)^2*M^4- 4*tan(theta)^2*M^2+ 8*tan(theta)^2)/M^2/(tan(theta)^2+ 1)/(- 8- 36*tan(theta)^4*M^4- 24*tan(theta)^4- 60*tan(theta)^4*M^4*gamma- 24*tan(theta)^4*M^2*gamma- 24*tan(theta)^4*M^4*gamma^2+ 8*M^6- 9*tan(theta)^4*M^6*gamma^3+ 6*tan(theta)^4*M^6+ 15*tan(theta)^2*M^6- 12*tan(theta)^6*M^2- 6*tan(theta)^6*M^4- tan(theta)^6*M^6- 12*tan(theta)^6*M^4*gamma- 12*tan(theta)^6*M^2*gamma- 6*tan(theta)^6*M^4*gamma^2- 9*tan(theta)^2*M^6*gamma^2+ 6*tan(theta)^2*M^6*gamma- 3*tan(theta)^6*M^6*gamma^2- tan(theta)^6*M^6*gamma^3- 3*tan(theta)^6*M^6*gamma- 12*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^6+ 3*tan(theta)^4*M^6*gamma+ 3*3^(1/2)*tan(theta)*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 3*3^(1/2)*tan(theta)^3*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 24*M^2- 24*M^4- 48*tan(theta)^2*M^4*gamma- 12*tan(theta)^2*M^2*gamma- 18*tan(theta)^2*M^4*gamma^2- 54*tan(theta)^2*M^4+ 36*tan(theta)^2*M^2- 24*tan(theta)^2)^(1/3)+ 12/M^2*(tan(theta)^2*M^2*gamma+ 2*tan(theta)^2+ 2+ tan(theta)^2*M^2+ M^2)/(tan(theta)^2+ 1))^(1/2)*(- 1/18*M^2*(6/M^2/(tan(theta)^2+ 1)*(- 8- 36*tan(theta)^4*M^4- 24*tan(theta)^4- 60*tan(theta)^4*M^4*gamma- 24*tan(theta)^4*M^2*gamma- 24*tan(theta)^4*M^4*gamma^2+ 8*M^6- 9*tan(theta)^4*M^6*gamma^3+ 6*tan(theta)^4*M^6+ 15*tan(theta)^2*M^6- 12*tan(theta)^6*M^2- 6*tan(theta)^6*M^4- tan(theta)^6*M^6- 12*tan(theta)^6*M^4*gamma- 12*tan(theta)^6*M^2*gamma- 6*tan(theta)^6*M^4*gamma^2- 9*tan(theta)^2*M^6*gamma^2+ 6*tan(theta)^2*M^6*gamma- 3*tan(theta)^6*M^6*gamma^2- tan(theta)^6*M^6*gamma^3- 3*tan(theta)^6*M^6*gamma- 12*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^6+ 3*tan(theta)^4*M^6*gamma+ 3*3^(1/2)*tan(theta)*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 3*3^(1/2)*tan(theta)^3*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 24*M^2- 24*M^4- 48*tan(theta)^2*M^4*gamma- 12*tan(theta)^2*M^2*gamma- 18*tan(theta)^2*M^4*gamma^2- 54*tan(theta)^2*M^4+ 36*tan(theta)^2*M^2- 24*tan(theta)^2)^(1/3)+ 6*(2*tan(theta)^4*M^4*gamma+ 4*tan(theta)^4*M^2*gamma+ tan(theta)^4*M^4*gamma^2+ 4*tan(theta)^4*M^2+ tan(theta)^4*M^4+ 4*tan(theta)^4- 8*M^2+ 4*M^4+ 4- 3*tan(theta)^2*M^4*gamma^2+ 2*tan(theta)^2*M^4*gamma+ 4*tan(theta)^2*M^2*gamma+ 5*tan(theta)^2*M^4- 4*tan(theta)^2*M^2+ 8*tan(theta)^2)/M^2/(tan(theta)^2+ 1)/(- 8- 36*tan(theta)^4*M^4- 24*tan(theta)^4- 60*tan(theta)^4*M^4*gamma- 24*tan(theta)^4*M^2*gamma- 24*tan(theta)^4*M^4*gamma^2+ 8*M^6- 9*tan(theta)^4*M^6*gamma^3+ 6*tan(theta)^4*M^6+ 15*tan(theta)^2*M^6- 12*tan(theta)^6*M^2- 6*tan(theta)^6*M^4- tan(theta)^6*M^6- 12*tan(theta)^6*M^4*gamma- 12*tan(theta)^6*M^2*gamma- 6*tan(theta)^6*M^4*gamma^2- 9*tan(theta)^2*M^6*gamma^2+ 6*tan(theta)^2*M^6*gamma- 3*tan(theta)^6*M^6*gamma^2- tan(theta)^6*M^6*gamma^3- 3*tan(theta)^6*M^6*gamma- 12*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^6+ 3*tan(theta)^4*M^6*gamma+ 3*3^(1/2)*tan(theta)*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 3*3^(1/2)*tan(theta)^3*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 24*M^2- 24*M^4- 48*tan(theta)^2*M^4*gamma- 12*tan(theta)^2*M^2*gamma- 18*tan(theta)^2*M^4*gamma^2- 54*tan(theta)^2*M^4+ 36*tan(theta)^2*M^2- 24*tan(theta)^2)^(1/3)+ 12/M^2*(tan(theta)^2*M^2*gamma+ 2*tan(theta)^2+ 2+ tan(theta)^2*M^2+ M^2)/(tan(theta)^2+ 1))+ M^2*gamma+ M^2+ 2)/(1/36*M^2*(6/M^2/(tan(theta)^2+ 1)*(- 8- 36*tan(theta)^4*M^4- 24*tan(theta)^4- 60*tan(theta)^4*M^4*gamma- 24*tan(theta)^4*M^2*gamma- 24*tan(theta)^4*M^4*gamma^2+ 8*M^6- 9*tan(theta)^4*M^6*gamma^3+ 6*tan(theta)^4*M^6+ 15*tan(theta)^2*M^6- 12*tan(theta)^6*M^2- 6*tan(theta)^6*M^4- tan(theta)^6*M^6- 12*tan(theta)^6*M^4*gamma- 12*tan(theta)^6*M^2*gamma- 6*tan(theta)^6*M^4*gamma^2- 9*tan(theta)^2*M^6*gamma^2+ 6*tan(theta)^2*M^6*gamma- 3*tan(theta)^6*M^6*gamma^2- tan(theta)^6*M^6*gamma^3- 3*tan(theta)^6*M^6*gamma- 12*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^6+ 3*tan(theta)^4*M^6*gamma+ 3*3^(1/2)*tan(theta)*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 3*3^(1/2)*tan(theta)^3*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 24*M^2- 24*M^4- 48*tan(theta)^2*M^4*gamma- 12*tan(theta)^2*M^2*gamma- 18*tan(theta)^2*M^4*gamma^2- 54*tan(theta)^2*M^4+ 36*tan(theta)^2*M^2- 24*tan(theta)^2)^(1/3)+ 6*(2*tan(theta)^4*M^4*gamma+ 4*tan(theta)^4*M^2*gamma+ tan(theta)^4*M^4*gamma^2+ 4*tan(theta)^4*M^2+ tan(theta)^4*M^4+ 4*tan(theta)^4- 8*M^2+ 4*M^4+ 4- 3*tan(theta)^2*M^4*gamma^2+ 2*tan(theta)^2*M^4*gamma+ 4*tan(theta)^2*M^2*gamma+ 5*tan(theta)^2*M^4- 4*tan(theta)^2*M^2+ 8*tan(theta)^2)/M^2/(tan(theta)^2+ 1)/(- 8- 36*tan(theta)^4*M^4- 24*tan(theta)^4- 60*tan(theta)^4*M^4*gamma- 24*tan(theta)^4*M^2*gamma- 24*tan(theta)^4*M^4*gamma^2+ 8*M^6- 9*tan(theta)^4*M^6*gamma^3+ 6*tan(theta)^4*M^6+ 15*tan(theta)^2*M^6- 12*tan(theta)^6*M^2- 6*tan(theta)^6*M^4- tan(theta)^6*M^6- 12*tan(theta)^6*M^4*gamma- 12*tan(theta)^6*M^2*gamma- 6*tan(theta)^6*M^4*gamma^2- 9*tan(theta)^2*M^6*gamma^2+ 6*tan(theta)^2*M^6*gamma- 3*tan(theta)^6*M^6*gamma^2- tan(theta)^6*M^6*gamma^3- 3*tan(theta)^6*M^6*gamma- 12*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^6+ 3*tan(theta)^4*M^6*gamma+ 3*3^(1/2)*tan(theta)*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 3*3^(1/2)*tan(theta)^3*(16+ 32*gamma+ 16*tan(theta)^4*M^2+ 16*tan(theta)^4+ 24*tan(theta)^4*M^4*gamma+ 64*tan(theta)^4*M^2*gamma+ 72*tan(theta)^4*M^4*gamma^2- 16*M^6+ 32*tan(theta)^4*M^6*gamma^3- 4*tan(theta)^4*M^6- 20*tan(theta)^2*M^6- 40*tan(theta)^2*M^6*gamma+ 8*tan(theta)^4*M^6*gamma^2- 8*tan(theta)^4*M^6*gamma+ 24*tan(theta)^4*M^4*gamma^4+ 72*tan(theta)^4*M^4*gamma^3+ 8*tan(theta)^2*M^4*gamma^4+ 40*tan(theta)^2*M^4*gamma^3+ 80*tan(theta)^4*M^2*gamma^2+ 32*tan(theta)^4*M^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^2+ 96*M^4*gamma- 48*M^2*gamma^2+ 32*tan(theta)^4*gamma+ 16*tan(theta)^4*gamma^2+ 32*tan(theta)^2*gamma^2+ 40*tan(theta)^2*M^6*gamma^3+ 5*tan(theta)^4*M^8*gamma^4- 6*M^8*tan(theta)^2*gamma^2- 16*M^6*gamma^2- 32*M^6*gamma- tan(theta)^4*M^8+ 64*tan(theta)^2*gamma+ 48*M^4*gamma^2- 4*M^8*tan(theta)^2*gamma- 5*tan(theta)^4*M^8*gamma^2- 4*tan(theta)^4*M^8*gamma+ 8*tan(theta)^4*M^6*gamma^5+ 4*tan(theta)^4*M^8*gamma^5- M^8*tan(theta)^2*gamma^4- 4*M^8*tan(theta)^2*gamma^3+ 32*tan(theta)^2*M^2*gamma^3+ tan(theta)^4*M^8*gamma^6+ 20*M^6*tan(theta)^2*gamma^4+ 28*tan(theta)^4*M^6*gamma^4- M^8*tan(theta)^2- 48*M^2- 96*M^2*gamma+ 16*gamma^2+ 48*M^4+ 120*tan(theta)^2*M^4*gamma- 32*tan(theta)^2*M^2*gamma+ 104*tan(theta)^2*M^4*gamma^2+ 48*tan(theta)^2*M^4- 32*tan(theta)^2*M^2+ 32*tan(theta)^2)^(1/2)*M^2+ 24*M^2- 24*M^4- 48*tan(theta)^2*M^4*gamma- 12*tan(theta)^2*M^2*gamma- 18*tan(theta)^2*M^4*gamma^2- 54*tan(theta)^2*M^4+ 36*tan(theta)^2*M^2- 24*tan(theta)^2)^(1/3)+ 12/M^2*(tan(theta)^2*M^2*gamma+ 2*tan(theta)^2+ 2+ tan(theta)^2*M^2+ M^2)/(tan(theta)^2+ 1))- 1))

Enjoy!
 
D

DorkBoy

Diamond Member
Sep 25, 2000
3,591
0
0
Originally posted by: Descartes
Originally posted by: DorkBoy
I'm a Nuclear Eng. but I don't remember sh*t
Click to expand...

That's comforting. I'm a doctor, but I don't remember anything either. I'm offering discounts for patients I get from AT.
Click to expand...

Hey Pal, I surf for a living.
And what kind of discounts? drugs or what?

 
R

RaynorWolfcastle

Diamond Member
Feb 8, 2001
8,968
16
81
Originally posted by: daveymark
ReynorWolfcastle, I didnt quite get that? Can you explain it one more time?
Click to expand...

There were 5 roots to the equation, you may have found one of the 4 others, would you like me to post them? ;)
 
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